What FAA regulations govern a US International Airport name change, such as McCarran International to Harry Reed International?
The owner of the airport determines the name.  If there is a dispute, courts have settled naming rights.

Whether the airport offers customs determines the "International" designation.  Some airports have no international flights, yet have the "International" in their name. They will have customs available, at least part time.

Side story: At one time, using a grass strip a couple of farms over, I asked a neighbor who worked for US Customs if I could fly directly to the grass strip (which did not even have a FAA Airport Identifier assigned) and clear customs there.  He said I could, and had me notify customs inbound, so they would notify him to meet me at the grass strip for inspection.  Doing that made trips from Canada to back home free of the hassles of dealing with a big airport.