My boyfriend is taking his son's mother to court to get visitation rights. He pays child support through the courts and gets 1hr a week supervised visits at the courts. He owes
Back arrears for child support, who gets that back child support money?

Back-support will likely go to the mother.

XenoSapien

Will depnend, is there money due the state for welfare that the child received, often that has to be paid back.

But remember back support is merely the support he did not pay when he should have. So it is money due and owed to the mother for the child.