Looking at this answer just now, I got to thinking about the writing on the back of his old Potions book.
There’s nothing to indicate that the school book in question did not start out as being simply Snape’s own book, bought at Flourish and Blotts like pretty much all new school books; so when he started using it in Potions class during his sixth year at Hogwarts, the book would have been new and have no one else’s scribbles and names in it. (Edit: Actually, as Au101 points out in the comments below, the date in the book is nearly 50 years before HBP is set, which is long before Snape’s time at Hogwarts; so this doesn’t seem to be true. I can recall no mention of any non-Prince scribbles in the book, though—or indeed of any previous owner writing their own name—so this probably doesn’t change much for the purposes of the present question.)
We also know Snape is relatively unpopular at Hogwarts, except (usually/sometimes) with Lily and (probably) with his Slytherin mates and fellow Death Eaters-to-be. A group of friends that tend to put quite a bit of stock in pure-bloodedness. As far as is generally speculated (though there is no canon either way to my knowledge), Snape never told any of his friends (except probably Lily who at least vaguely knew his parents) that he was half-blood.
It seems quite safe to assume that Snape would have had Potions together with at least the other Slytherins in his year, and presumably also with students from one or more of the other houses.
It also seems quite safe to assume that, as in the real world, Hogwarts students occasionally share books, borrow each others’ books, get each others’ books mixed up by accident, or in some other way happen to look at/in other students’ school books. That certainly happened to me many a time when I was in school.
Perhaps exactly for this reason, it’s common practice for students to write their name in their books, to show whose book it is. Snape here chose to write “This book is the property of the Half-Blood Prince” on the back of his book, rather than simply putting his name on it.
Considering the kind of home-made spells he jotted down in that book, it’s perhaps not entirely surprising that he didn’t want his own name on it for any teacher to read. But he could have thought of a better nickname for himself—one that didn’t give away his half-blood status, for example.
There would be a fairly substantial risk of one of his Death Eater-to-be mates at some point during the year (or years, if the same book is used in the seventh year) coming across this inscription of ownership one way or another. (S)he would probably wonder, and presumably also ask Snape, what in the blazes he meant by calling himself a ‘half-blood prince’—which would at the very least lead to some awkwardness in that particular circle of friends, and at worst possibly even leave Snape ousted completely, and essentially entirely friendless.
Running a risk like that seems quite out of character for someone who’s usually as cautious, withheld, and calculating as Snape is portrayed as being.
So why did Snape write that his Potion book belonged to the ‘Half-Blood Prince’ for anyone in class to see? Why not either think of another, less incriminating, nickname; or not commit the nickname to writing?
There’s definitely no canon answer for this in the books. Has Rowling ever mentioned it? Has it ever been mentioned in any of her other writings? I’m looking for in-universe answers, primarily canon ones if possible. Well-argued speculation is all right, too, if there is nothing in canon (as I suspect there isn’t).