Patron has the benefit of tradition. It also acknowledges that financial support comes those who use the library, but not in a payment for service model.
Customer reminds everyone on both sides of the desk that the library is there to serve its users. Without its customers, the library would go out of business.
Members emphasizes the community aspect of library services, and may instill a sense of ownership amongst its ‘card-carrying’ members.
Users takes a ‘just the facts’ approach. It also promotes a sense of the library as a system that needs a positive overall user experience, beyond good customer service from the staff.
Other possibilities include client (common in corporate and other special libraries), visitors, guests, (both of which emphasize the need for hospitality), and many less common terms.
Which is your favorite?