As far as I am aware, the foot in football originally referred to the fact that it was a game played on foot rather than on horse back (as opposed to polo for example).
I actually looked into this recently. It's related to standard naming conventions established by England. Originally there was Association Football (soccer), gridiron football (American football) and rugby football (obvi rugby). Regarding American football,
"Both soccer-style football and rugby-style football eventually found their way to America. What resulted was an American combination of the two games. It wasn't until much later (1906) that forward passing was allowed. So because the American game was really just another form of the European football games, it too became known as football."
Looking it up, football is a class of games where kicking is involved in scoring points. There's a bunch of sports that fall under the category, Association Football (soccer), various flavors of Rugby and various flavors of Gridiron Football (which uses kicking for field goals after scoring a touchdown)