I predict that it will be revealed that Quirrell or a closely related entity has been abusing Harry on and off throughout his life, to try and make him into a Dark Lord.

He can go to Harry's house like the time he played Father Christmas.

Obliviated memories leave residue, which is how in Chapter 88 the twins remembered that they could find people, in the castle, but couldn't remember how.

In the first chapter, Harry noticed that he believed in magic.

some part of Harry was utterly convinced that magic was real

In chapter 16, Harry is almost reminded of something when he looks at Quirell, but can't remember what. And when Quirrell is first introduced, Harry ominously recognizes him

"Professor?" Harry said, once they were in the courtyard. He had meant to ask what was going on, but oddly found himself asking an entirely different question instead. "Who was that pale man, by the corner? The man with the twitching eye?"

"Hm?" said Professor McGonagall, sounding a bit surprised; perhaps she hadn't expected that question either. "That was Professor Quirinus Quirrell. He'll be teaching Defence Against the Dark Arts this year at Hogwarts."

"I had the strangest feeling that I knew him..." Harry rubbed his forehead. "And that I shouldn't ought to shake his hand." Like meeting someone who had been a friend, once, before something went drastically wrong... that wasn't really it at all, but Harry couldn't find words.

In the sixth chapter, McGonagall points out that Harry can act like an abused child.

sometimes, you say or do something that seems very much like... someone who spent his first eleven years locked in a cellar.

Quirrell uses Obliviation and memory charms and as Mr. Cloak-and-Hat, he manipulated Blaise. And he uses Obliviation and memory charms more subtly, to change someone's mood and personality over time, as shown when he brute-force-save-scumed his way to making Hermione suspicious of Draco.

Quirrell expected Harry to become a Dark Lord when he spoke with him after the first class and was surprised that Harry aspired to science.

Quirrell expects the worst out of people, and so he expected that an abused Harry would be destined to darkness.

Edit: I just realized that Harry was probably abused almost every night (or day) for some significant period. There was a time turner involved, and that's why his sleep cycle is off.