Professor Horace Slughorn: he's fat, has a taste for crystalized pineapple, and was potions professor to Voldemort, Snape, Lily, and Harry; to name a few. We all know his importance to the series; he holds the memory that proves that Tom Riddle knew about horcruxes and almost certainly proves that he made at least one.
But does Slughorn have another, more sub textual, purpose?
"Good Lord, it’s clear you’ve inherited your mother’s talent, she was a dab hand at Potions, Lily was!"
This quote, taken at face value, is little more than an offhand comment about a brilliant potioneer. But is that all?
Lily Potter was not known for being clever: popular with inheritable traits maybe, but never clever. When paired with a boy rather good at rule breaking, wouldn't you expect their child to be smart (no offense, Harry, but do you really think Snape would believe your nickname is "Roonil Wazlib"?)
That brings me to the first part of my argument. Severus Snape was a very clever young man with a special proclivity for the dark arts. His brilliance was, more than likely, inherited from his mother; who had an award in the trophy case at Hogwarts.
Brilliant as he was, why did Snape bother writing down all of his little tricks and hints to make potion making easier? These contrast starkly to the homemade spells that he used his potions book to write down. Did he really need them?
Lily was his first, and only, real friend in the wizarding world; and I believe those scribbled notes were for her. Remember how pensieve Lily said that her friends couldn't understand why she was spending so much of her time with Snape? Wouldn't it be reasonable to assume that some of that time be in a potions class? Wouldn't Snape, who had seen his friend struggling time and time again, resolve to help his only friend?
Lily, who "broke-up" with Snape around her sixth year, probably would have kept the book out of a necessity to keep up her reputation as a successful potions student. I can't imagine Snape would have minded; he was infatuated with her for the rest of his life. If anything, I am certain that Snape would have implored her to keep the book.
When Lily left Hogwarts, she probably would have donated her old potions textbook to the school; she wouldn't need it in the Order.
Which brings me to the second part of my argument. In all the time that Snape was potions master at Hogwarts he never removed his book from the store-cupboard; which begs the question of whether he even knew it was there? Being the slimy, sniveling piece of work that he was, why would he leave students a clear path to potions excellence?
If Snape ever actually allowed a student to borrow a textbook from the store-cupboard (taking away points is much more fun), would a student ever tell Snape that their textbook gave him or her an unfair advantage over other students? And so would said student turn this book in? If you're simply using a book with helpful notes on it, why turn it in?
When Slughorn took over from Snape, it was the first time in the series that we actually read about students borrowing potions books, so I don't believe that Snape knew his book was there.
When Harry grudgingly picked up the book, it may have been the first time the book was touched since his mother put it away. Slughorn's offhand comment about Lily being a dab hand at potions could have had much more significance than we thought.