I'd just ask her. You say she's been very private about her situation - that makes me think she wouldn't want to have the pink ribbons as a constant reminder of what might lay ahead, but she may say that it would be a comforting reminder of what she's been able to put behind her.
If you'd prefer to keep the quilt a secret for the time being, then I'd probably use something else for the backing - something that either coordinates with the colors used in the top or something that is more reflective of her life (horses for a horse-lover, mountains for an out-doorsy person, coffee mugs for a coffee lover, etc.).
I hope you'll come back and share pictures when you're finished!