I was playing hearthfire and my daughter softie walked up and asked if she could keep a rabbit she found, but it wasn't there? If I tell her to keep it, what will happen?

Do it and find out.

I'd imagine that would have been quicker than starting a thread and waiting for an answer.

A rabbit? That must be the one I always forget. It will likely appear should you say yes. They're not always in plain sight.

I did say yes and nothing appeared.

I reloaded a save, I was just wondering If it was going to stop her from having another pet.

You adopted child will continue to bring one of several pets home until you agree to let him/her keep one.

See http://www.uesp.net/wiki/Skyrim:Adoption for a full guide.

Invisible pet has been reported as a known bug. It occurs where the home has a cellar and the creature glitches/gets stuck on the lower level. It can only be rectified if your adopted child enters the cellar/same cell as the creature. As you have done, the other option is to reload a previous save.

I hope you don't get some invisible beast that keeps growling around the house, like the snow fox... At least you could kill the fox...