In HBP, Harry finds the old potions book and uses it to good effect meaning that the advice makes sense. If it were truly Snape’s original thoughts in there then it could mean that he was helping Lily Evans all throughout potions classes and that’s why Slughorn kept remarking how gifted she was in potions. BUT Slughorn does tell Snape that Harry gave him the perfect Draught of Living Death. And says, “I’m not even sure if you, Severus.” Or something to that effect. That makes us think that maybe Snape wasn’t gifted in Potions to begin with.
That leaves us with the possibility that maybe Lily Evans was helping Snape throughout potions classes. Maybe a lot of the advice written down is from Lily. Hermione even reckons that some of the writing is girly. Maybe Lily even wrote some of them in the book. That would mean that Harry is actually getting help in Potions from his mom which is a nice thought.
We also know that Severus applied for DATDA position first and settled for potions when he didn’t get the job. Maybe he wasn’t as keen on potions as it seems. That would also explain why Snape kept the book for so long if it reminded him of Lily and he even used it as a bit of a journal it seems.