I am not sure I understand what the conflict is or who would be adversely affected by it, so I would start by asking her to cite a document that spells out the conflict and asking her what she wants done about it. 

If your organization has a conflict of interest policy, it could have a provision requiring a director to recuse in deciding whether to make a grant to an organization at which a relative is employed. But even if it does, and even if the relationship is close enough to be called a conflict, the policy would normally require only recusal from a vote and not resignation from the board.  (See Ready Refence Page: “Conflict of Interest Policies Help Avoid Problems”).

The county library system may also have a conflict policy, but it could probably not impose conflict rules on you as an independent organization.  The system’s policy could possibly say the relative could not be involved in administering relationships with your organization, but it is unlikely that it could cost the relative their job because your organization supports the library where the relative works.  Just because a librarian has a rich uncle who wants to support the library should not mean that the librarian should lose their job.  Rules would not ordinarily be that stupid.  

I am not sure what the new director is demanding, but if she can’t show a clear basis for it and she persists in demanding it, I would raise the issue with members of the board that appointed her.  They should be concerned about any policy that discourages support for a local branch of the library system and may be able to suggest that the new director back off gracefully.