A rich kid who (or their family) is always donating new books to the library - possible twist that they can't stand reading but constantly have to go to the library for said donations or are giving away books they keep recieving but don't want.
A couple of star-crossed sweethearts who exchange messages through the books they borrow. (Could be harmless post-it notes or actually scribbling in the books themselves)
Could have a kid that starts out as a loner, visiting the library every day during lunch breaks and whatnot, but slowly over the course of the year they gain friends and you only see them when they pop in for group study sessions with said friends.