I was going to suggest the same as elain... except with the addition of natural food coloring or a tiny bit of juice to tint the whipped cream! (I bet a dash of beat juice could make a pretty pink)
Regular frosting will be sticky and there is a very high chance the cake will end up in the babies hair! Keep is simply to wash out so that you don't have to worry about it and can have a fun time!
Take plenty of photos and go ahead and let the baby make a mess! If her clothes will be "nice" or fancy, switch to a plain onesie or easy outfit right before the cake! Try to put a plastic tarp or tablecloth under her highchair for the cake too- and just let her at it!
If you want a healthy cake, consider carrot cake, or many healthy "muffin" recipes could be baked into a cake! I also think that angel food cake is usually pretty healthy, especially if you make it from scratch...
Happy 1st Birthday to your Daughter!
-M.
Oh- this site has some awesome recipe ideas for baby's first birthday cake:
http://www.wholesomebabyfood.com/tipFirstBDay.htm