It's All About Dudley
Dumbledores last and most important letter to Petunia Dursley must have been quite some piece of manipulation to convince the woman to accept Harry as a surrogate son (Chapter 37, OotP). Dumbledore is all about choices; he would be unlikely to stoop to blackmail, bribery, or trickery to manipulate Petunia into accepting Harry. For example, I doubt that the act of carrying the baby over the threshold tricked Petunia into some kind of magical contract. Dumbledore is above such things.
No, I think Dumbledore basically told Petunia the truth, gave her some very hard choices to make, and hoped for the best. Thats what he does. To paraphrase Mad-Eye Moody, he is a trusting kind of guy. But Dumbledore is also an able manipulator, Im sure he laid things out in a way that really left Petunia no option but to agree.
Jo Rowling has told us that other, prior, letters were sent to Petunia Dursley from Albus Dumbledore. One can realistically assume that this was a two-way communication, with Petunia perhaps repeatedly writing back saying (shrieking?), No, I wont have him! I imagine that it is primarily from these correspondences that Dumbledore forms the opinion that Petunia only grudgingly, furiously, unwillingly, bitterly (Chapter 37, OotP) accepted Harry.
Of course, the earlier letters might have been about other matters. Perhaps they shared a passion for ten-pin bowling? Perhaps Dumbledore was writing to Petunia for several months filling her in on the antics of those crazy Death Eaters and forewarning her that, yeah, youre going to need to take in your nephew if your sister happens to die in an act of sacrificial love that causes a death curse to rebound on Lord Voldemort thereby vanquishing his physical body but leaving his evil soul at large and you as her blood sister are the only candidate for providing a lasting sanctuary for the child
No, I tend to think that the letters themselves are way simpler than people think. I think Dumbledore exchanged several letters with Petunia after Lilys death but before delivering Harry, with each one seeing an escalation of panic on Petunias side and gravity on Dumbledores.
I can imagine Dumbledore starting by writing a heartfelt and moving letter to Petunia about the Potters deaths and a simple plea for Petunia to accept Harry into her home. Petunia is startled and responds more or less politely but firmly, no, thank you very much (and she very much hopes that that is the last of it).
Dumbledore tries again, appealing to her sense of duty and obligation to her sister. Petunia starts to lose it and begins to rant and rave. Dumbledore persists, Petunia resists. Fawkes is going back and forth like some phoenix-feathered tennis ball. Finally, Dumbledore is forced to hit her where it really hurts.
So, what convinced Petunia Dursley to accept a wizard into her home? What could Dumbledore possibly offer to sweeten the deal if family loyalty was not enough?
I think its all about Dudley.
The reason could be enormously complicated and far-fetched (such as binding Dudleys magical powers if he happened to turn out to be a wizard), but I dont think it needs to be. I think it comes down to two simple and understandable fears attributed in canon to Petunia and Vernon Dursley.
1. That people would discover their connection to the magical world; and
2. The safety of their precious son, Dudley.
The first fear is pretty straightforward, the second - not so much. Why would Petunia be afraid for her son? After all, the Dursleys had apparently had little contact with the magical world in several years (Chapter 1, PS/SS). And we know that Petunia Dursley was aware of Lord Voldemort being vanquished at some point in the past when she says, [hes] back? (Chapter 2, OotP). So, when corresponding with Dumbledore and saying no, I wont have him, what was it that Dumbledore wrote to convince Petunia that a serious risk to Harry and/or her own family remained?
I believe that Dumbledore told Petunia that Lord Voldemort was not really gone, that some day he would return, and that immediate threats also existed from amongst the dark wizards legion of angry supporters. In The Lost Prophecy chapter (Chapter 37, OotP), Dumbledore told Harry:
You were in more danger than perhaps anyone but I realized. Voldemort had been vanquished hours before, but his supporters - and many of them are almost as terrible as he - were still at large, angry, desperate, and violent. (Chapter 37, OotP)
I believe that Dumbledore would have expressed all of these fears to Petunia. He would have been telling the truth if he warned Petunia that her family was also at risk by virtue of her connection to Lily Potter. At this stage, the Longbottoms had probably not yet been attacked, but a similar vengeful pursuit for information from the Dursleys would definitely have been a realistic fear at the time.
But such alarming thoughts could only increase Petunias resistance to the notion of opening her home to Harry Potter. So, what was it that Dumbledore was able to offer to quell this fear and turn Petunias decision around (even if bitterly, furiously, grudgingly)?
The manner of Lilys death provided Dumbledore with an opportunity to create a lasting sanctuary against Lord Voldemort. This would protect Harry, but would Petunia care enough about her nephew to overcome her fears for her own familys safety and privacy? Somehow I think not (and thats an evil on a scale I dont even want to think about). So why then did Petunia risk all and accept Harry into her home?
I think its all about Petunias surrogacy of Harry having a side benefit of also protecting her own child as well, a child who also shares Lilys blood. I think its significant that Rowling has never referred to Dudley as a blood relative of either Harry or Lily. I think this omission is deliberate, and that its meant to trigger an oh yeah kind of moment for the reader when wondering why Petunia would give a toss if Harry is protected or not.
As for the letters themselves, Ive written up a one-shot fic containing sample letters to illustrate how things might have gone.
Canon evidence behind this theory
In Chapter 1 of PS/SS we are told in authorial voice very early in the chapter that:
Mrs Potter was Mrs Dursley's sister, but they hadn't met for several years
Even so, this does not mean that letters had not been exchanged. Both Vernon and Petunia knew that Lily had had a baby boy called Harry who was about the same age as Dudley. Lily and James would have made arrangements for Harrys upbringing (should something happen to them), but they would have been unlikely to choose Petunia and Vernon. It would have much more likely been Sirius or other families in the Order like the Longbottoms or Bones. There would have been no particular motivation then, in trying to involve Petunia (in advance) in plans for Harrys care just in case anything happened to the Potters.
No one could have predicted that baby Harry would have managed to vanquish Lord Voldemort in a twisted scenario that left his parents dead and his Aunt Petunia as his number one option for a surrogate mother.
In Chapter 1, we observe some important things through Vernon Dursleys point of view. Firstly, there are an awful lot of owls flying about (perhaps some might have been going to/from Petunia?). Then during the day at work Vernon overhears things, and contemplates ringing his wife, but he decides that:
There was no point in worrying Mrs Dursley; she always got so upset at any mention of her sister.
When Vernon arrives home he finds that:
Mrs Dursley had had a nice, normal day. She told him over dinner all about Mrs Next Door's problems with her daughter and how Dudley had learned a new word (Won't!).
Between arriving home and going to sleep, Vernon broaches the subject:
He cleared his throat nervously. "Er - Petunia, dear - you haven't heard from your sister lately, have you?"
As he had expected, Mrs Dursley looked shocked and angry. After all, they normally pretended she didn't have a sister.
"No," she said sharply. "Why?"
Note that Petunia answers no to a question about having heard from her sister. There is nothing to say that she hasnt been getting letters from Dumbledore all day.
"Funny stuff on the news," Mr Dursley mumbled. "Owls... shooting stars... and there were a lot of funny-looking people in town today..."
"So?" snapped Mrs Dursley.
"Well, I just thought... maybe... it was something to do with... you know... her crowd."
Mrs Dursley sipped her tea through pursed lips. Mr Dursley wondered whether he dared tell her he'd heard the name "Potter. He decided he didn't dare. Instead he said, as casually as he could, "Their son - he'd be about Dudley's age now, wouldn't he?"
"I suppose so," said Mrs Dursley stiffly.
"What's his name again? Howard, isn't it?"
"Harry . Nasty, common name, if you ask me."
"Oh, yes," said Mr. Dursley, his heart sinking horribly. "Yes, I quite agree."
Vernon doesnt say another word on the subject and Petunia is the first to fall asleep that night. Petunia wouldnt have dropped off to sleep so easily if she was expecting a baby to be dropped on her doorstep at midnight. First conclusion: Petunia did not know that Harry was coming that night.
My conjecture is that earlier in the day, Petunia received news of her sisters death and a plea from Dumbledore for Harry to be taken into her home, but that the last reply she sent back to Dumbledore was a very firm NO! NO! NO! and shes hoping that those horrid people have given up on the idea. Her behavior then, when Vernon arrives home, could be interpreted as trying to act normally and cover up - something she tries so very hard to do given that her greatest fear is to have her connection to magic discovered.
In Chapter 37, The Lost Prophecy, in OotP, Harry is told many important things, but Ill just highlight a couple here:
Dumbledore tells Harry that he guessed when he saw his scar that a connection [had been] forged between you and Voldemort.
He tells Harry that:
Five years ago you arrived at Hogwarts, Harry, safe and whole, as I had planned and intended. Well - not quite whole. You had suffered. I knew you would when I left you on your aunt and uncle's doorstep. I knew I was condemning you to ten dark and difficult years.
Dumbledore admits to pre-knowledge of Harry being consigned to a difficult home life before he left him on the doorstep. He also intended for Harry to be safe and whole. I choose to think his intentions were honorable; he just had really poor options to work with.
my priority was to keep you alive. You were in more danger than perhaps anyone but I realized. Voldemort had been vanquished hours before, but his supporters - and many of them are almost as terrible as he - were still at large, angry, desperate and violent.
Dumbledore identifies his priorities - safety for Harry from both Lord Voldemort and his supporters. It is likely that the sanctuary charm offers a type of general protection, not just against Lord Voldemort, but also against any mortal danger. We also learn from Lord Voldemort himself (in GoF) that Harry was protected even better than the boy knew. If you recall, too, when Vernon Dursley attempts to strangle Harry in Chapter 1 of OotP the man suddenly yelps, as though he had received an electric shock, and releases Harry - this could be natural wizarding self-protection or it could be part of the sanctuary charm - who knows?
Anyway, Dumbledore goes on to explain why it had to be Petunia.
But I knew, too, where Voldemort was weak.
I put my trust, therefore, in your mother's blood. I delivered you to her sister, her only remaining relative.
She doesnt love me, said Harry at once. She doesn't give a damn -
But she took you, Dumbledore cut across him. She may have taken you grudgingly, furiously, unwillingly, bitterly, yet still she took you, and in doing so, she sealed the charm I placed upon you. Your mother's sacrifice made the bond of blood the strongest shield I could give you.
I was always very curious as to how Dumbledore knew this if he didnt have additional communication with Petunia (i.e., more letters or visits either before or after he dumped Harry on her doorstep). Think about it, Petunia isnt going to betray grudging, furious, bitter behaviors in public towards a tiny toddler - irritable and long suffering, sure, but not furious and bitter - shed be too worried about what neighbors like that new Mrs. Figg would think!
By the way, this initially moderate behavior would have also been necessary for Harry to develop speech and other social skills. Harry couldnt be the normally functional ten-year-old child that we first see in Book 1 if he was comprehensively isolated and maltreated from the age of 15 months. So my conjecture is that Dumbledores anticipation of Petunias behavior as grudging, furious, and bitter when she took Harry must have been a combination of letter correspondence at the time plus, possibly, things Lily might have told her friends about her sister and brother-in-law.
Her blood became your refuge. You need return there only once a year, but as long as you can still call it home, whilst you are there he cannot hurt you. Your aunt knows this. I explained what I had done in the letter I left, with you, on her doorstep. She knows that allowing you house room may well have kept you alive for the past fifteen years.
Harry figures out that Dumbledore sent the Howler.
I thought, said Dumbledore, inclining his head slightly, that she might need reminding of the pact she had sealed by taking you. I suspected the Dementor attack might have awoken her to the dangers of having you as a surrogate son.
It did, said Harry quietly. Well - my uncle more than her. He wanted to chuck me out, but after the Howler came she - she said I had to stay.
Note that Dumbledore makes no mention of any kind of blackmail - no inference that something bad will happen to Petunia if she breaks the pact. This is a highly significant point. Weve heard of binding magical contracts like when the names come out of the Goblet of Fire, but I suspect that Petunia didnt actually enter into some kind of unbreakable contract. If she did, then no Howler would be necessary; merely a quiet little note saying, Er, Petunia? Dear? You really have no choice in this matter.
Dumbledore is big on choice, and I think that Petunia is probably free to choose whether to keep allowing Harry houseroom or not. So, there must be something in it for Petunia, thats the way deals work. I give you something; you give me something in return.
In Chapter 2, A Peck of Owls, in the OotP, Harry reveals that Lord Voldemort is back and a huge change comes over Petunia.
Back? whispered Aunt Petunia.
...Her large, pale eyes (so unlike her sister's) were not narrowed in dislike or anger, they were wide and fearful. The furious pretence that Aunt Petunia had maintained all Harry's life - that there was no magic and no world other than the world she inhabited with Uncle Vernon - seemed to have fallen away.
Yes, Harry said, talking directly to Aunt Petunia now. He came back a month ago. I saw him.
Her hands found Dudley's massive leather-clad shoulders and clutched them.
Petunias reaction to hearing that Voldemort is back is to clutch at Dudleys shoulders. She is frightened for her son, pretty simple, yes, but why? Why would Voldemort pose any threat to Dudley (or herself or Vernon)?
Vernon, on the other hand, is not at all impressed by any of this; he wants to chuck Harry out but then Dumbledores Howler arrives:
REMEMBER MY LAST, PETUNIA.
Aunt Petunia looked as though she might faint. She sank into the chair beside Dudley, her face in her hands.
...The boy - the boy will have to stay, Vernon, she said weakly.
If we throw him out, the neighbors will talk, she said. She was rapidly regaining her usual brisk, snappish manner, though she was still very pale. They'll ask awkward questions, theyll want to know where hes gone. We'll have to keep him.
My supposition is that Petunia perceives a threat to her own son because of warnings Dumbledore issued in his last letter, and that she believes that by honoring her agreement to keep Harry, then she will maintain some kind of protection over members of her own family.
Petunias rapid shift from wide-eyed fear to her usual brisk, snappish manner (after announcing that Harry must stay) forcibly reminded me of her behavior way back in Chapter 1 of PS/SS. Her behavior with Vernon that November evening (before finding Harry on her doorstep) is stiff and evasive, she looks shocked and angry, she snaps, she purses her lips, she answers questions ambiguously.
My conjecture is that prior to finding Harry on her doorstep, she has been in communication with Dumbledore on the topic, that she knows what has happened to Lily and James, but believes that she has been successful in saying no to Dumbledore about accepting Harry into her home.
How very wrong she was