"My Fair Lady" a pun?

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?

"My fair lady" comes from the children's song: "London Bridge is falling down, falling down, falling down, / London Bridge is falling down, my fair lay-dee."