For one I went to years ago they gave everybody a cute little favor as they came in, a tiny triangle folded cloth diaper closed with a safety pin. When the time for the games came around, everyone looked inside their diaper and the one with the poop (a dot of mustard) was the winner.
At another one the game was trying to guess what the mystery baby food was in each of several jars of baby food with the labels removed. The first round was sniff test only. For the second round tiny ice cream sample spoons were provided for taste testing.
Another fun game was trying to come up with as many names as possible for each letter of the alphabet that nobody else had come up with. These could either be ones you really liked or ones that you wouldn't name a baby on a bet. The latter option can really be hilarious!