Speaking of Slughorn, my wife pointed out this: in HBP, Slughorn keeps saying that Harry seems to be a natural at potions, just like his mother was. Now that we know from book 7 that Lily was Severus' best friend at Hogwarts, perhaps she was such a "natural" at potions because she was getting help just like Harry was, from Severus (she directly, Harry from the HBP's potion book.)
Makes me wonder if Slughorn knew what was happening all along, and was secretly being sarcastic when he kept praising Harry...
Makes sense.
I think perhaps Slughorn intentionally gave that particular book to Harry. We know that Slughorn is a name-dropper ... who better to have in your debt than The Boy Who Lived? Not to mention that the Boy in question is the son of one of your favorite students from years gone by?
If not for Snape, Harry might have been quite the dab hand at Potions.