Did you tell her you were going to check references after you gave her the offer? Or did you have the references, but just not get around to checking them beforehand?
I ask because if you said it as you were giving her the offer, then I would assume if I were her that your offer was contingent on good references. You can now go back and say that you thought the family member would have more substantial information about her work reliability, and that without other references, you're sorry, but you'll have to move on.
Otherwise, it will be less graceful, but you'll just have to say what you say here: on further reflection, you just can't hire someone with no work history to work in your home with your child. You're sure she's a wonderful person, but you need some sort of work history to go on since you won't be there to supervise, and you hope she'll find something soon.
It's 10/hour a week job; I suspect she'll be bummed, but it's not like you're pulling a full time nanny gig.