CoS: Harry finds out he's a Parselmouth and people think he might be the heir of Slytherin. Snape looks at him rather calculating after the incident with talking to the snake, like he had underestimated Harry.
HBP: Harry finds out he can make potions afterall (to Snape's surprise) and Slughorn claims he's as good at potions as Lily was... even supposedly better than Snape. Snape says he has underestimated Harry after the Sectumsempra incident.
CoS: Harry's tailing/spying on Draco because he thinks Draco has something to do with the opening of the CoS. Later finds out Lucius was behind planting the book.
HBP: Harry's tailing Draco because he's sure Draco is up to something, finds out Draco was behind the opal necklace, the poisoned sherry (or was it mead?), and behind letting DE's into the castle. This is Draco's punishment for Lucius's screwups....
CoS: Harry destroys the diary horcrux, not knowing that's what it was.
HBP: Harry finds a fake horcrux. Is the real one (the locket horcrux) still intact or has it already been destroyed?
The Parallels between books 3 and 7 could also be quite strong....
PoA: Harry (and Hermione) travels back in time trying to save at least one life...saves Sirius, Buckbeak, and himself.
Book 7: Harry (possibly with Ron and Hermione, but he'll probably go into the house by himself) travels back in time, trying to save his parents? He saves neither parent, but the baby Harry, of course, survives and gets that scar.
PoA: Harry thinks he sees his father conjuring a Patronus but later realizes it's a future Harry he's seeing. That allows him to conjur the Patronus when he needs to, since he's seen himself doing it.
Book 7: Baby Harry hears a noise but can't determine who's making it. Pensieve Harry thinks he's seeing his father under the Invisibility cloak, but then realizes it's a future Harry? This prompts Harry to do exactly what he saw himself do in the memory?
PoA: Harry leaves the cloak where Snape finds and uses it in order to sneak into the Shrieking shack.
Book 7: James happens to leave the cloak out (lying around the house somewhere) where Harry finds it in order to hide and watch the attack in the Potter house?
PoA: Everyone thinks Sirius Black was the spy and is in league with Voldemort, but he's innocent. Sirius, however, feels partially to blame for their deaths (James and Lily). Wormtail turns out to be the traitor but now owes a Life Debt to Harry.
Book 7: Everyone knows Snape is a spy and thinks he's in league with Voldemort, but/and ____? Is there another traitor? Will Snape prove to be a traitor afterall, or are we to see him redeem himself at the last moment? Snape feels partially to blame for their deaths (James and Lily). Wormtail repays his Life Debt to Harry.
PoA: When Harry confronts Sirius in the Shrieking Shack he temporarily forgets that he's wandless and he's underage. In his anger (based on his love for friends and family) He rushes toward Sirius and attacks him bare-handed.
In OotP we finally get to read the prophesy and the wording says "hand", not wand.
Book 7: What if, during the final confrontation with Voldemort, Harry ends up wandless (or he decides not to use it, since they have brother wands) and then...he forgets he's "weaker", forgets that it would be stupid to physically attack Voldemort...and he attacks Voldemort with his bare hands. Harry actually kills Voldemort without a wand...?
And of course:
PoA: Wormtail fulfills Trelawney's prophesy about the servant returning to his master.
Book 7: Harry and Voldemort fulfill Trelawney's prophesy about them.