There is a sign for "ROSE" if you would like to simply name her Rose. It's an easy enough sign for toddlers to do, although they might not do the "R" handshape perfectly at first. Even if it takes them a little while to get the handshape down, a two-finger tap on each side of the nose would be easy to recognize as the dog's name by the whole family.
Sign for ROSE: https://www.handspeak.com/word/search/index.php?id=4660