My 1 year old loves animals of all kinds... she likes to grab them by their ears and/or whiskers an squeeze and yank and pull and not too many animals like that... I wouldn't dare let her get ahold of a rabbit, guinea pig, chinchilla, bird, hamster, etc. My parents cats and dog know what she wants to do, and while one of the cats will put up with it for a few minutes, they tend to keep their distance. I have a few friends who have masochistic cats that actually seem to ~enjoy~ being yanked and pulled on, but you have to look out for the affectionate claws that go with cats like that...
My daughter's favorite "toy"? We've got a 20 gallon aquarium with fish and a small turtle (please do your research on turtles if you decide to go with a turtle... these are ~NOT~ pets for your child to handle until I'd say at least 5 years old, but if you take care of it properly, it will be alive at least until she's in her 30's so that's plenty of time for interaction. Also please be aware of adult size for any turtle you may get... anyway) It took her a while to notice the turtle anyway, but the fish have ~always~ been one of her favorite things in the house... just make sure it's up high enough that she can't get into it.
One is a bit young for actual pets... better to visit other people's pets than subject an animal to having to live with a little person who will just torment it, regardless of intentions.