I don't know about the cricket origin and the presentation of a hat (though it seems plausible enough), but as I understand it, in football (Association Football that is) in the early 19th century days it would have meant that when a player scored three goals in the course of a game, the hat would be passed round among the spectators so they could make a contribution.

Footballers being ordinary working blokes in those days, as was the case up into the 1960s, that kind of cash bonus would have mattered.