I was once again inspired to write this from an essay I read on the subject (one of those 'damn, they got there before me' kind of things)...
I was always very interested in the Advanced Potions book in HBP, even before we all found out that the book belonged to Eileen Prince and passed down to her son, Severus Snape. It struck me as odd, while I was in the middle of the book, that an obviously very precious book, full of invented (usually potentially dangerous) spells and amendments to potions, was just left with the spare supply in the Potions Dungeon. But I thought 'perhaps this Half-Blood Prince is careless, who knows, perhaps he/she died, perhaps he/she was arrested...' and didn't dwell much on why the book fell into Harry's hands.
But then, of course, we find out that it is Snape who is the Half-Blood Prince. And this confused me greatly.
Since when has Snape just left precious items belonging to him scattered around the place? Since when has Snape been even remotely careless and absentminded? He wouldn't be alive if he had possessed an inattentive streak - organisation is not only a part of his character, it is vital to his survival. He's a spy, for pete's sake. It's not fun and games.
Do I honestly believe that Snape of all people would just leave this book containing many dark curses, his inventions, just lying around in the cupboard of the potions class? Of course I don't.
The title of Potions Master is handed over to Slughorn in Harry's sixth year, this we all know. My guess is that Snape had spent a good couple of hours at least searching every single orifice of his class and putting everything into order before handing over the reigns. He obviously knew that his Potions book lay in the spare storage... was he wating for anyone to pick it up? I think not. I think the book fell into the hands of whom it was intended.
There are two differing opinions here, which I'm not really sure on... the first option is that BOTH Dumbledore and Snape planned it so that Harry would find his old schoolbook, however, even though this seems to be the view of the person writing the essay, I feel apprehensive to believe this. For one, the book is full of dark magic, and I don't believe that Dumbledore would agree to having Harry exposed to such spells as the Sectumsempra. The second option, which I am more inclined to agree with, is that this was purely Snape's doing.
Why would Snape leave his book for Harry to find? Well, if you recall sixth year and Harry's sudden surge of potency in potions, can we be in any doubt as to why? Harry has never been spectacular at potions, though in the sixth book with the Half-Blood Prince's aid - he becomes the star pupil. If we take the book away, how far would Harry have gotten? He would have been average at most. Potions is vital in the wizarding world - think of the potion surrounding the locket. Voldemort uses them extensively, the most prevalent being the rebirth in GOF. Without Snape's presence, Harry is average & feels calmer; but think about it... with Snape's indirect presence - Harry is spectacular.
We have Snape as the Defense Against the Dark Arts professor - another bonus for our scar-clad hero - and we also have Snape as his remedial Potions professor without... well... actually teaching the entire class. Considering Snape's extensive expertees in both subjects, Harry recieved the best intensive training he could get. And by the looks of it, it was through Snape's intentions that he did so.
Take a look at the Sectumsempra incident in the bathroom. The essayist pointed this out and I very much agree that two things initally struck me as odd in this scene - the first that Snape had lost his usually extremely excellent restraint of his emotions, and the second that he sent Harry to fetch the aforementioned book instead of actually following him every step of the way to get it. How many times in the books has Snape "lost it"? Not many. A couple at most. The first in the Shrieking Shack, and again in the hospital wing (I admit POA was not Snape's moment to shine) and then in Harry's fifth year during the Worst Memory scene. When Harry uses Sectumsempra on Draco we often blame Snape's outburst on the Unbreakable vow and I am in no doubt that it was partly responsible; but the anger could also be directed at the fact that Harry is deliberately misusing the curses that resides in Snape's book. There are specific reasons for Harry owning that book.
But that's really splitting hairs - what I was most interested in was the fact that Snape waited in the bathroom for Harry to get the book, instead of following him to make damn well sure that Harry wouldn't swap around the books. I don't believe for one second that Snape is stupid enough to assume Harry would not do such a thing. Harry also found the time to hide the Half-Blood Prince's copy, and while Snape does indeed call Harry a liar and gives him detention for the 'Roonil Wazlib' copy - he does not actively make any effort at all to find his old Potions book, where he could have done, and could have done without much effort.
There's probably a lot more examples, and I'll add them when they come to mind (usually in the middle of night - it's difficult to not think about Snape when you're in my room, trust me)... but to me, there is just no way that Snape left his book their unintentionally, whether Dumbledore was in on it or not. It was the book that pushed Harry ahead, beyond all others, and considering it is my mission to redeem Severus Snape and his actions to every single Potter fan out there... this is a comforting thought indeed.