I was thinking mayonnaise because it is, basically, eggs and oil (and salt, vinegar...), but in reading further I read that the egg was a leavening agent and found this sub:
2 Tbs. water
1 Tbs. oil
½ tsp. baking powder
I haven't eaten any of the cake, but tasted the batter tasted great and it rose well, so I'm thinking it was an ok thing to do, lol!