What is the ruling on the vacation salary that was wrongfully disbursed to my mother, and she was unable to return it to the responsible authority? Is it permissible for her to keep it, or to give it to her unemployed son?

If the responsible authority has been informed of the truth of the matter, then you have fulfilled your duty. If the responsible authority does not act, you must return the amount yourself if possible; otherwise, act as an agent to deliver the amount to those who are entitled to it among the poor and the needy, or for public interests such as building hospitals and schools. As for giving to an unemployed son, it is permissible if he is poor and in need to meet his needs, because he has a greater right to charity than other poor people.