A temporary bonus would be reasonable, and is even supported in the rules:
The time to make checks using a library, however, increases to 1d4 hours. Particularly complete libraries might even grant a bonus on Knowledge checks in the fields that they cover.
… but I'd lean away from giving any permanent numerical bonus (in particular for something the character is doing while the other PCs have to sleep) in favour of narrative effects that spotlight the character and allow them to bring something cool to the table.
If they get a success on a knowledge roll then, it could be narrated as "Oh, you remember reading something about Owlbears in the library last week, they are deadly hunters and their young are incredibly valuable."
You can use the same technique to infodump knowledge without a roll. If you're planning this, then you could create a handout in advance and give it to the player so they can relate it to the rest of the party themselves.
You can use the library as a source of quests. "While digging through a tome about the life of the secretive Archmage Zagreus, you come across some scribbled notes in a margin that look like they might be a guide to finding one of the entrances to his pandimensional fortress".
Other people use the library, and they might need or be able to offer help to the party in some way. Dave points out that you can represent this mechanically using NPC Boons (at the time of writing, I can't access paizo.com but the content is also available via the wayback machine).
The PC will probably acquire a reputation for spending so much time at the library, so much so that people wanting to get in touch might visit the library explicitly to do so. This can lead to scenes set in reading rooms and quiet corners of an atmospheric building, with occasional shushing from librarians. It would make a great change from the usual three venues of Heroes Doing Business (the tavern, the private study and the throne room).
While the rules for levelling do not demand justification, it also provides an opportunity for the player to explain (in the fiction) why they allocate their skill points and spell acquisition to particular places when they level up.