As I follow it, you go from the 1500s Italian game "mallet ball," or "pallamaglio," to the French "pallemaille," to the English name of the game, "pall-mall," to an area where you would play pall mall being called a "mall," to a walking area that you could play pall mall being called a mall, to an indoor shopping area where you walk from store to store being called a mall, to a person who hangs out at the shopping area being a mall rat—the term "mall" all coming from the Italian and then French term for a mallet used in an outdoor game several hundred years ago.