Is it permissible to give an impoverished sister from Zakat money to pay off her son's debt, considering his inability to do so, in order to spare her family and legal troubles?

Your nephew is not liable unless he transgresses or is negligent. If liability is established against him due to his transgression or negligence, or if he was compelled to pay the money to ward off harm from himself, then it is permissible for you to pay your Zakat to settle his debt. And your paying Zakat to him is more deserving than paying it to a stranger, for charity given to a relative is both charity and strengthening of kin ties.