Who claims the child as a deduction has no bearing on who has physical custody, if someone's paying support as ordered. Usually the decision is made based on what's best for everyone, and sometimes it's part of the court decision by the judge (based on demands or recommendations from the two sides).
If her ex is in the hole for support, claiming the child as a dependent on his tax return doesn't erase his obligation. It does not say "The child lives with dad so dad no longer has to pay mom." It MIGHT give him a little more money in his pocket but there has to be a guarantee that SHE gets that money! If she gets more in her pocket from child support (particularly if his wages are garnished), it could mean more for her than the deduction means on her tax return. In which case it's to her benefit.
But I have no idea why anyone thinks that the father taking the child as a deduction will "clear his slate with child support" - it doesn't mean he won't owe for back support, and it doesn't mean he won't have to pay going forward. Without knowing more, I'd say that she shouldn't give him anything unless she's got a guarantee of it coming back to her as a benefit. If he's ducking support now, I don't see how this will change. But she should consult her own tax prep accountant or her divorce attorney for the best advice in her particular situation.