How about a button to get a random opponent and color?
Would work like this:
1. Click on the button and your "invite" would go into a hidden pool.
2. If there is an opponent in the pool of hidden invites, you get the game and the color is assigned randomly.
3. If there is no opponent in the pool waiting, the next opponent who clicks on the button gets you from the pool.
There could be selectable rating ranges and timebanks.