Jack, as a mayor would simply be a mayor in the US, I was thinking about different countries, I could believe that a Dutch Afrikaner I knew would call a woman mayor a "Lady Mayor" and I thought possibly in Italy that a woman mayor would be a "Mayoress".
For the US I would say Mayor is the correct term... until the Politically Correct crowd shows up and decides that Lady Mayor is derogatory as is suggests a subset of Mayors, settling on Mayoress as the PC term for a Mayor who happens to be female. Personally I would use Mayor and the rest is gobbledygook.
Well for me I agree but these Ladies Still like that Segregation, just as one off we still call our Lady Police officers WPC ( Woman Police Constable One day we will get Equality anything from the Continent (Mainland Europe? )