Jo Rowling has already explained what Dumbledore was talking about in his howler, Remember my last! On her website she explains what that letter was so I wont reiterate it here. I would like to set forth my theory of what Petunias mystery is -- actually what she is: a muggle, a witch, or a squib.
Is Aunt Petunia a Squib?
Good question. No, she is not, but-[Laughter]. No, she is not a Squib. She is a Muggle, but-[Laughter]. You will have to read the other books. You might have got the impression that there is a little bit more to Aunt Petunia than meets the eye, and you will find out what it is. She is not a squib, although that is a very good guess. Oh, I am giving a lot away here. I am being shockingly indiscreet.
So far we only know of three types of people in the HP universe: wizards/witches, muggles, and squibs. We then have some facts given by JK -- and then some fuzzy action. First: She is not a Squib. Second: She is a Muggle, but-. Third: She didnt mention "witch" anywhere in there. What is that but- doing in there?
Apparently JK is giving something away here, and no one seems to have picked up on it. I have a theory about a part of an element in the HP universe that has already been introduced, and then its implications on one of our more (ahem) lovable characters in the HP series, Petunia Dursley. What JK is giving away is that Petunia is a Muggle, but- and I think this implies that Petunia is a witch. Hear me out! Stop! No wait!
My theory deals with choice. JK has demonstrated in her writing of the HP books what a big element choice is, and I predict that she will reveal another surprise part of that element of choice: Wizards and witches who have chosen to opt out of being what they were born to be.
What do we as readers know about each of these types of humans?
Wizard/witch: There is no definition of a wizard or a witch in the books except for vague assumptions. We know they are magical and can make funny, unexplainable things happen subconsciously.
Squib: (According to Ron) "A Squib is someone who was born into a wizarding family but hasnt got any powers." (CoS)
Muggle: (According to Hagrid) "A Muggle," said Hagrid, "its what we call nonmagic folk like them." (PS/SS)
Now lets look at the options and choices. Does each of these three types of humans we know of have a choice of being what they are? Although our initial question was whether a wizard/witch could choose, lets look at everyones choices. Logically speaking:
A muggle usually doesnt know that there is an option of wizardry so they dont have much to choose from. (What else is there?)
A squib knows about muggles and wizards/witches. A squib can either choose to live as a squib with the magic world (Filch) or as a muggle (Mrs. Figg). So basically a squib can choose to be a squib or a muggle.
Wizards and witches have the most options available:
--A wizard/witch can be a wizard/witch.
--A wizard/witch can choose to be squib living in the wizarding world (according to their faith in themselves, that is -- Neville until he started gaining confidence in himself and kicking butt).
--A wizard/witch can choose to be a muggle who has little or no contact with the wizarding world. (What happens to all those magical children that decide magic just isnt for them? Who says being a wizard/witch is the best path to take?)
Other than the evidence of logic, the HP books show us a place where this theory of wizards/witches being able to choose is possible, at least in Vernons and Petunias eyes:
We swore when we took him in wed put a stop to all that rubbish, said Uncle Vernon, swore wed stamp it out of him! Wizard indeed!
If he wants ter go, a great muggle like you wont stop him, growled Hagrid. (PS/SS, bolding added)
The Durselys thought they could stamp it out of him and even Hagrid admits that if Harry wanted to go, he could. But what if he didnt want to? Lets not look too narrowly at a decision! What if there are children from muggle families that dont want to go learn magic for whatever reason? There is the possibility!
So not only according to logic but according to these quotes from the books, there is such a possibility of a wizard/witch not wanting to be a wizard/witch but rather, a muggle. I would assume that a child getting the opportunity of playing with magic would only turn it down in weird circumstances. This brings us to the implications of this theory on Petunia.
I think JK is hiding that Aunt Petunia is actually a witch who chose to be a muggle. JK wouldnt outright lie to us -- but she would play with our minds -- hence the but-! What else could she possibly be giving away here?
Now that is established, heres where I think Petunia fits in. A passage giving a lot of valuable insight on Petunia:
"Knew!" shrieked Aunt Petunia suddenly. "Knew! Of course we knew! How could you not be, my dratted sister being what she was? Oh, she got a letter just like that and disappeared off to that--that school--and came home every vacation with her pockets full of frog spawn, turning teacups into rats. I was the only one who saw her for what she was--a freak! But my mother and father, oh no, it was Lily this and Lily that, they were proud of having a witch in the family!
She stopped to draw a deep breath and then went on ranting on. It seemed she had been wanting to say all this for years. (PS/SS)
What can we determine from this? Aside from the obvious--Petunia really doesnt like her sister Lily, not now and not then, and Petunia has quite a lot of built up rage--what strikes most important is that Petunia felt ignored by her parents (as Lily got all of the attention) and that Petunia seemed surprised that her parents were proud to have a witch in the family.
These last two points, that Petunia felt ignored, and that she was surprised about her parents' reactions, are what give her away. I see three scenarios. Was Petunia or Lily the older sibling? Who received their letter first? Or was Petunia not qualified enough to go to Hogwarts (a Stan Shunpike perhaps?):
1) The most plausible scenario there is, is this: Petunia was the oldest sibling, received her letter, didnt like the sound of being a witch and thought her parents would be outraged anyway. So she rejected the letter without her family finding out. Then, Lily receives her letter sometime in the near future and accepts. Much to the surprise of Petunia, her parents are actually pleased! Petunia, never liking the idea from the start, is hurt that her parents cant see how terrible being a witch is, making her bitter and jealous of the attention Lily gets.
2) Lily received her letter, then Petunia received hers as the younger of the two. Once seeing all of the attention Lily received from being accepted, you would think Petunia would want to jump onto the wagon. The only reasoning I see in this possibility is that Petunia was simply the black sheep of the family and wanted to rebel, and did not accept the invitation.
3) Petunia is a witch, but not invited to Hogwarts. We as readers dont know if Stan Shunpike wasnt invited, or if he simply wanted to go into a trade. If it is possible for a witch to not be invited to Hogwarts because they arent magical enough, then Petunia could fall into that category. The evidence in this is that in the passage above, her bitterness may come from jealousy.
Why would Petunia be so against the wizarding world in the first place? Two classic examples to explain this:
1) When a person is a part of a religion, if that person rejects that religion, he or she is usually extremely bitter against that particular religion, where as if a person isn't part of that religion in the first place, he or she is simply apathetic. Petunia obviously hates the wizarding world--everything about it. Perhaps she tasted it and rejected it.
2) A person has an illness but refuses to accept it and is in denial. This person naturally dislikes anybody who has and is accepting of the same illness--it is threatening to that person who is so sure there is nothing wrong with them. In other words, a person who is in denial is to dislike/hate everyone who is not in denial. Being a witch is not an illness but Petunia definitely seems to think so. But why?
She is very concerned with what other people might say. She and Vernon have both demonstrated how much they prioritize being normal and living in a way so people around them will not notice anything strange about them. I happen to think that their constant struggle to be un-noticed is not all a product of Harry. I think Petunia has always been afraid of what other people might think of her. This would stem from her childhood and the funny occurrences she knew she caused. She knew she was different being a witch. She was probably very aware of others feeling about her, and exaggerated them in her own mind. So of course when she received a letter confirming her own abnormalness, she was distraught, thinking it was a terrible thing--an illness. She never could accept magic, and she could never accept herself, which led to bitterness, jealously, a loathing of the wizarding world, and a constant fear that she might be discovered.
One more piece of evidence is the Creevy brothers--both are wizards and muggle-born. It is therefore possible and seemingly expected that if one muggle-born is a wizard, his or her siblings are as well.
I really do wonder what JK is talking about there! This is my best guess--and I hope you made it through with me! I highly doubt Aunt Petunia will have much affect on the plot of the next two books, but it would explain why she is the way she is, and open up a new way to look at people in the HP universe.