What is the ruling of Islamic law on giving the usurious money obtained from a usurious bank—of which my mother was unaware of its prohibition until a long time later—to the poor suitor as a form of assistance to him?

The remaining interest-based (riba) gains must be disposed of and spent on Muslim welfare or given to the poor and needy. It is permissible to give it to your fiancé if he is poor, and it will then be lawful for him. Your mother must stop depositing money in interest-based banks, withdraw her money, and invest it in Islamic banks.