I just bought myself a Deering Goodtime banjo. It's my third banjo, and I'm wishing it would have been my first. It really does live up to the hype and it would have been the perfect banjo for me to start out with. It's lightweight, has awesome tone, and since it's not all dolled up, I don't mind if it gets a bump or two (which is sure to happen with a kid).
I'm guessing you're thinking the parlor banjo since it's shorter? I think that's a great idea, though not necessary if you want to save a little money and just go with the good ol' Goodtime orginal. The 9-year old kid from Sleepy Man Banjo Boys plays the full size with no problems. It's just whatever you get used to.
Kudos for encouraging and supporting your kid in this endeavor! It'll make memories with her dad that she'll never forget.