Twenty minutes spent actively engaging each other sets the tone for any meeting of strangers. A game with simple rules and no more objective than to entertain will make coffee time more relaxing. When the game engages the group in activity that can't be completed without the group then there is another lesson learned.
Examples
Toss a ball from person to person by calling the receiver's name before the toss. Follow the same path until everyone learns one other's name. Introduce more balls until congestion happens. Then, with no more instruction, ask everyone to move so as to optimize the ball motion. Team discovers the circle and laughs at how much better it works.