First you should be sure that your child does not have wheat or egg allergy. I think a plain sponge cake with vanilla icing/ fruits is the safest bet, but having said that eating too much of it won't obviously be good for a 1 year old. Personally I would find it a hassle cleaning up after she messes up an entire cake. Also the risk of the cake getting in her eyes/ nose/ ears. I gave my daughter little bit of chocolate/ strawberry cake which we got for her 1st birthday and she was fine with it
i made a simple cheesecake made of cream cheese, eggs, vanilla beans & added a TBS of sugar for my daughter when she turned 1, not only did she love it (carved out her name on the cheesecake & she gobbled almost 2 out of the 5 alphabets of her name). i made a small pan for mess sake & a different pan for proper eating for adults.
What about cupcakes? We stuck a 1 year old candle in one and the bub had her own 'cake' - she ate probably one quarter of. You can get baby size ones or regular that actually come in the shape of a real cake. Make your own or some companies do custom cakes so you could request specific/organic/natural ingredients.
My daughter just turned one last weekend and I made a tiny carrot cake for her (halved the sugar). She really wasn't sure at first but then had a blast taking fistfuls and eating some and smearing the rest.
Another source for healthier cake recipes (low sugar) is Ruth Yaron's Super Baby Food book.