Are there FAA rules about changing an airport name from Municipal to Regional?
In general there are no strict rules on airport names. This would be more of a marketing decision, and it's up to the owners/operators to decide of the expense of the "rebranding" is worth the benefits.

US regulations do cover international airports, and the Secretary of the Treasury designates the official list of airports. Even then, there are plenty of international airports not on this list, and there are even some airports called "international" with no customs facilities or plans to have them in the near future.