Warning: Spoilers below regarding content on the site and questions from the Sorting quiz.
The Sorting Hat is one of the cleverest enchanted objects most witches and wizards will ever meet. It literally contains the intelligence of the four founders, can speak (through a rip near its brim) and is skilled at Legilimency, which enables it to look into the wearer's head and divine his or her capabilities or mood. It can even respond to the thoughts of the wearer. (J.K. Rowling, Pottermore)
Other than this short section from Pottermore, (and a paragraph about "hatstalls," when the Sorting Hat takes longer than five minutes to make a decision), Jo doesn't give us a whole lot of new information about this perplexing magical object - though she does give some fans the chance to try it on. Perhaps by looking at some of the questions the Pottermore Sorting Hat put to beta-users we can draw a few more conclusions about how it operates.
In no particular order, here are a few out of the 200 questions beta-users encountered on the Pottermore Sorting quiz, the possible answers, followed by a brief analysis.
After you have died, what would you most like people to do when they hear your name?
1) Miss you, but smile.
2) Ask for stories about your adventures.
3) Think in admiration about your achievements.
4) I don't care what happens when I'm dead, it's when I'm alive that matters.
While there were other questions like this one with four possible answers, beta-users quickly learned that this wasn't your usual four answers-four houses (no-brainer!) kind of test. One could argue that choice 3 is one that a Slytherin would pick, considering their ambition to rise above the rest, but then again a Ravenclaw who is centered on his/her academic prowess might also pick it, too. Certainly if Gryffindors are most noted for their bravery, it follows that they enjoy it when the events of their lives become the stuff of school legends... couldn't a Gryffindor also select choice 3? Obviously it would depend on the individual, but this first question from the Hat reflects on the true complexity of the whole batch.
Which would you rather be?
This question gave me particular trouble - I swayed between Liked and Trusted. Given the choices though, (and my knowledge of the book series), I think I can assert with relative certainty choice 4 would be a solidly Slytherin pick. (Not saying Slytherins are all evil, but how many Hufflepuffs do you know that most desire to be feared?) There might be a shot that a Ravenclaw would pick Feared, but only if they were immensely competitive in their academic studies. Similarly I think a Gryffindor might also consider choice 4 - as she would want enemies to shrink at her valor - but would likely select another choice since she would not want to scare her Hufflepuff fans. Imitated, Envied, Liked and Praised as choices all seem to possess a certain vanity, connected to a person's projected image as perceived by their peers. Gryffindors, Slytherins and Ravenclaws would float between these most likely, depending on the individual, though I'd be willing to bet that most Ravenclaws would go for Praised. Trusted seems to be the obvious choice for Hufflepuffs, considering their strict loyalty, but there is a shadow of deceit within the very word - to desire to be trusted is to desire to be the carrier/receptacle of another person's secrets. It is a submissive, yet powerful position, which is why I think it might also be the choice of some Slytherins willing to betray their friends if it meant climbing the ranks.
Which of the following would you most like to study?
This one is purely a matter of personal preference, but as we know with everything Jo does, every answer here must weigh towards a single house or collection of houses. Centaurs are perhaps the easiest to place - as they are creatures naturally curious about the universe and the way it works, it's not a large leap to say that they are most closely aligned with the base ideology of Ravenclaw. Choice 1 seems special, however, as it reflects more on a person's want to study ghosts than the ghosts themselves. Desire to study ghosts is a desire to study Death, an attempt to pry into its inner workings before it can get you, or your loved ones. We know of a few characters that are obsessed with this kind of thinking - Harry, Dumbledore... we could say that Gryffindors, then, would go for ghosts; (Death being that last thing to conquer for those natural born fighters) but remember Luna also showed special interest in the veil just as Harry did in the Department of Mysteries. It's true that she was connected to Harry in this (as she also lost a mother) but matters of life and death are also large, universal questions, which I'm sure would intrigue many a Ravenclaw.
Goblins, trolls, and merpeople are rougher creatures which have had difficulty being incorporated into Wizarding society over the centuries. With the exception of goblins, who as far as we know hold at least a few other common jobs beside running and operating banks, these creatures have generally been subjugated by humans in the present and have therefore become romantically attached to their primal pasts, both idealized ones and real. For the excitement that the challenge offered, Gryffindors would likely choose to study trolls, specifically because these are the most dangerous of the above creatures. Admiring their cunning, Slytherins would be drawn to Goblins, and the two houses would be about equally split on merpeople. Slytherins would also likely choose to study Vampires, the darkest of magic creatures, also because they are the ultimate loners, bent on getting exactly what they want when they want it - the very embodiment of a bloodthirsty Slytherin.
The werewolf is an interesting choice on the list. While certainly one of the most dangerous, it is also one of the most vulnerable of magical monsters. It is one part human and one part animal, and therefore possessor of all the qualities the human species does not wish to associate with itself - violence, irrationality, great desire... I believe a Hufflepuff would be sympathetic to this creature's struggle, a hybrid of two worlds yet outcast of both. The Hufflepuff (while scared) would really just want to make the werewolf happy.
Heads or tails?
Some beta-users were, reasonably, frustrated when they got this question. Seemingly we were being asked to make a random choice and that was going to affect which house we were sorted into - a clever move on the part of Jo, I first decided, as an element of chance must truly affect all of the Hat's decisions.
But is this exactly true? What if certain personality types are attracted to only one or the other? One could argue choosing Heads, for instance, signifies a dominant type of character, who isn't afraid of leading while Tails implies a type of personality that is slightly unsure, or doesn't feel like making as much of an impact. Thus saying, we might decide that Gryffindors and Ravenclaws would pick Heads and Hufflepuffs and Slytherins Tails but this one, truly, might just cater to the individual.
Stars or Moon?
This one may be slightly less arbitrary than the heads or tails question. In our culture, the stars and the moon have taken on symbolic meanings as they have related to humankind over the millennia. Very generally, the moon is closely tied to the unconscious of humanity - as the "lesser light" in the sky as compared with our Sun, (and reflector of that light), the moon has become literarily synonymous with the western notion of Eve, from the Bible's Genesis, and therefore the feminine, and therefore (rather unfortunately) physical desire. The stars, on the other hand, have always proved a vast mystery to man, though he has desperately sought to map them out, and since gaining knowledge of the heavens has culturally been linked with the highest of intellectual endeavors this has become a (symbolically) masculine practice.
Given all that, it seems Ravenclaws would be naturally inclined to choose Stars because, being that they reach higher than the influence of the moon, they reflect that highest knowledge of the universe. Gryffindors, too, seem to be living in these larger terms, and the stars also speak to a higher justice in Nature (outside of earthly, human justice) that Griffindors would likely be attracted to. Slytherins are naturally connected to their desires, so they would generally feel a stronger bond to the moon - which is closer and more tied to representations of (and the goings on of) humanity. Odd enough, Hufflepuffs would also be aligned with the moon because they lack the desire to move further then what they know and are exposed to. They are simply far to content.
Considering these questions, what have we learned? It seems to me that the Sorting Hat has two yous in mind when he sorts you - the person you truly are and the person you project. If we consider the old case of Harry Potter when he first tried on the Hat, we actually see this duality of consciousness in action:
"Hmm," said a small voice in his ear. "Difficult. Very difficult. Plenty of courage, I see. Not a bad mind either. There's talent, oh my goodness, yes - and a nice thirst to prove yourself, now that's interesting....So where shall I put you?"
Harry gripped the edges of the stool and thought, Not Slytherin, not Slytherin.
"Not Slytherin, eh?" said the small voice. "Are you sure? You could be great, you know, it's all here in your head, and Slytherin will help you on the way to greatness, no doubt about that - no? Well if you're sure - better be GRYFFINDOR!"
(Sorcererâs Stone p. 121)
Though within him Harry had the makings for a good Slytherin (by nature, yes, not by Voldemort) Harry chose Gryffindor because he was acting according to his projected personality, the model of what he wants to become - a person on the side of justice, a loyal friend and, if perhaps not a celebrity, someone fairly well-known. That is why, at the end of the day, the Sorting Hat is a misnomer because we are ultimately the ones doing the sorting.