What is the ruling on redressing an injustice on behalf of a relative who participated in a theft as a child without his knowledge or being informed, and is he absolved by that, and is what the questioner did—returning the money on his own behalf without the owner's knowledge—valid, and is the questioner sinful for not informing his relative, and is it permissible to pay off a debt on behalf of someone else without their knowledge, and what if the owner of the money dies?

What you did by returning the stolen money is obligatory upon you, and you are not required to inform the owner of the money about the true situation. There is no sin upon your friend if the act occurred before he reached puberty, but this stolen money is a debt upon him that he must repay. It is permissible for you to repay the debt on his behalf, even without his permission or knowledge, and in doing so, you are a benefactor who will be rewarded. If the rightful owner dies, the right is to be paid to his heirs.