If you're not going to have too many people coming, just have it at your house... that way your DD can play with her toys & take a nap if she needs to. We had a BBQ/party... the kids played outside & the adults hung out too. You can buy a cake with your baby's favorite animal on it or something, but she's just 1, so she won't care about any of it. Don't forget to take pictures. :)
My daughter just had her ist birthday.. I just had it outside in the yard.. I filled the baby pool, Ran the sprinkler and the slip n' slide and the kids had a Great time! Even a few of the adults went on the slip n' slide and we all had a very good time!
Home is fine, maybe even best. There will be extra people around, different things going on, that will stress your baby. Being at home will be one familiar thing. Remember, the party is really for you and the other adults.