It's often fun to play best 2 of 3, although if you don't ask for a rematch quickly, the opponent is gone before you can ask.
How about adding a button after you've played a game where you would select 'rematch' or 'best 2 of 3'?
There are different ways this could be implemented, but I believe it's best to make the evaluation after you've already played one game.
A slick way to program this would be for the button to normally say 'Request rematch' but if you'd already split 2 games evenly, the label changes to 'Best 2 of 3?'
I bet someone could program this in a couple of hours. Easy-peasy and would add a fun dimension.