At six years old, one question you really want to be asking is, "what kind of animal do you or your mother want to be taking care of?" Because it's going to come down to an adult to ensure that whatever it is is healthy, happy, and otherwise well-cared for.
After you figure out what you and/or your mom want to take care of, then start thinking about something that goes with her personality and abilities -- for example, a rat might be nice, but what about when she's "playing with" the rat and the rat decides it wants to run? Will she always have the rat-room secured so it doesn't end up holed up in the underside of a stove (that happened to a co-worker of mine). Or can you or you mom always be with her to supervise?
I was thinking maybe if your mom is interested in hamsters, a guinea pig might be nice. They're larger, harder to "lose," are often both interactive and snuggly.
Oh... but what about the mom's cat vs rat/hamster/piggie situation? Predator and prey...