When in yeshiva, a friend (who I think is now a rabbi) told me that when he once went to return an overdue book to the library, the librarian told him that she was waiving the overdue fee. He had then argued that she had no right to waive the fee and that he wanted to pay it. She wasn't going to argue about it so accepted it. Was this friend correct that the librarian had no right to waive the fee?
According to R. Yisroel Belsky, no. Employees are given a little leeway in giving customers extra or charging them less, in order to encourage the customer's use of the facilities. Specifically in regard to libraries, R. Belsky writes:
It could be that the librarian may feel that you're such an excellent customer that he'll waive the fee... Still, if you feel that the person is causing a loss to the library, you should pay the fine and not accept the favor.The same applies to waiters who give you a few extra french fries or charge you a little less in order to create a pleasant atmosphere. Big favor, however, might be theft.
Favors are not something that's outside of proper behavior in a business context. When limited to areas that create good will, the small amounts of favors that are done are not only proper and acceptable, but they're chiyuvim (requirements) as well.