We would have to rotate who gets to pick, because there is no way on EARTH we could ever agree on anything.
It might be interesting if we were tasked with finding an obscure game in an area (be it a system, a developer, an era, a genre, etc) that we are deemed an "expert" in. We could open the floor to "who should pick, and under what parameter."
For example, vudu could suggest the following: "I want Jon to pick any game from the Hoop and Stick Entertainment System, since that's what he grew up with."