"My Fair Lady" a pun?

Posted by Word Camel on January 07, 2002

Does anyone know why the musical is called "My Fair Lady"? It struck me that "My Fair" could be a joke based on the cockney pronounciation of "Mayfair". And while we are at it, I thought "cockney" specifically referred to East end accents. If that's the case, what are accents in other parts of London referred to as?