It looks like only Citi ($1k available) and Merrick ($600 available) actually have enough available on them to even consider a balance transfer? If those have very low or 0% balance transfers available, and she's paying a ridiculous interest rate on other cards, then it might make financial sense to transfer those and save the interest. Given that utilization though, I don't think you're going to help her score much by doing anything other than paying down the balances.
I'd attack cards with the highest interest first. If some of the small balance cards have very high interest, then maybe try to zero those with balance transfers. That might help her score a tiny bit, just by decreasing the number of accounts with balances, and it could save some money on interest and let you pay them down a little faster.
If you do decide to balance transfer though, be very careful with the timing. I've had balance transfers take up to almost a month to fully post. The worst case scenario is that the receiving card reports with the balance transfer already on it, but the transferring card doesn't post the transfer until after it reports, and so you can end up with a month where BOTH cards are including the same balance on your credit report. That might not be the end of the world, but it also might make creditors nervous.
If you do pay off most/all of the cards, that should shoot her score WAY up, assuming no other negatives. I'd guess then she could probably get some nice CLIs. At that point, you'd probably want to look at possibly closing some of the less desirable cards, like First Premier, and heading for prime cards like Amex, Chase, Citi, etc.
