Is it permissible to give charity from Zakat funds on behalf of a deceased son whose debt owner is unknown, especially given that this is the sole source of income?

If the mother confirms that the paper is in her son's handwriting and indicates his acknowledgment of a debt, it is considered acceptable evidence. If the creditor cannot be found after searching, she should give the amount in charity on his behalf. Then, if he (or his heirs) is found later, they are given the choice between approving the charity or reclaiming the money. As for settling the debt from charity funds, there is no harm in that, as the recipient of charity owns it and has the right to dispose of it.