As other posters have said, it doesn't really matter what kind of cat/dog she has (and if no allergies she'll be healthier for it in the long run). Pet hair isn't going to be getting into her lungs and clogging them up. I guess theoretically it sounds like something that could happen but that just isn't how our lungs work. Most of our breathing is done through our nose and the hairs in our nose are there to trap anything that might get in there (dust particles, pollen, etc). Pet hair doesn't float around the house in the air. It is "heavy" and falls to the floor. Even if baby is on the floor she isn't going to be sucking up pet hairs. I have had 2 crawling babies and neither one of them has choked on pet hair <img src="i/expressions/face-icon-small-smile.gif" border="0"> My little 13 month old loves giving the pets hugs (which often means a face smooshed against all that hair). I am more worried about the carpet and the dirt/fibers than the pets.
My opinion (so take it for what it is worth) - save your money and get a pet from a shelter who needs a home. We have lots of cats (mostly strays who have adopted us), 2 small dogs, 3 chickens and some fish. I have never lived w/o pets and Jessica is right - the studies are showing that it is GOOD for children to have pets - helps build our immune systems and reduces the risk for asthma and allergies. And the emotional benefit to having a pet is huge and should not be understated.
I guess the other question - are you up for more work? Cats require little maintenance but they still are more work than not having anything. And they can rack up some serious vet bills (we just dropped about $300 this weekend for one of ours). Not to mention litter box cleaning if they don't go outside.