What should a husband do with his wife who asks for a divorce and waives her rights, then returns and demands them in full?

It is not permissible for a woman to seek divorce without a legitimate reason. If she does so, she is sinful, and the husband is not obligated to divorce her. He has the right to refuse to divorce her until she ransoms herself from him with money or the like. Divorce by way of ibra' (absolution) waives the agreed-upon financial rights owed by the husband. Reconciliation is advised first, and divorce should be the last resort when its benefit outweighs other considerations.