We use a recipe similar (not identical, but similar) to the above poster's recipe for my egg-allergic daughter's cake:
Dry Ingredients:
1 1/2 cups flour
1 cup sugar
1/2 tsp salt
1/4 cup cocoa
1 tsp baking soda
Wet Ingredients:
1 Tbsp. vinegar
1/3 cup oil
1 tsp. vanilla
1 cup cold water
Mix and pour into ungreased 8 inch round or square pan (if you're going to make stacked cakes, put parchment paper or wax paper in the bottom of the pans so you can easily turn the cakes out of the pans). Bake 350 degrees until done (about 35 minutes). This can also be made into cupcakes. You can double this up and will have two rounds (what we usually do for birthday cakes).
It's a good idea to let the mom know you'll be doing this, and to give her a copy of the recipe you use, so that she can feel comfortable with what you've done. I often bring something 'just in case' for my dd, and we talk about how she might not be able to have the cake so she isn't too sad if the cake isn't safe.
Buttercream frosting (homemade) will work fine to frost. Royal icing and some of the 'decorations' for the cakes have egg, so do check those before decorating the cake.

As a mom of an egg-allergic kiddo, thanks for thinking of this and being inclusive. Sometimes I just bake cupcakes and bring them with me to supplement whatever the birthday kid is serving, because it really is sad for my dd when she can't have the cake. It's nice when people are inclusive like this.
