If the relative about whom you are asking is entitled to zakaah because she is poor and in need, or because she is in debt, then there is no reason why you should not give her zakaah, rather it is better to give it to her, because it is both zakaah and upholding ties of kinship. See the answer to question no. 20278.
But you should advise her not to enable her son to spend it on haraam things, rather she should buy him whatever he needs of permissible things. She can hide the money from him and spend it on herself and her dependents.
You should also advise the son and call him to Allaah; do not despair of his being guided, for people’s hearts are between two of the fingers of the Most Merciful, and He turns them as He wills. How often has He accepted the repentance of a sinner and how often has He guided one who was astray.
If it is not possible to hide the money from the son and prevent him from spending it on haraam things, then the mother can take the money and give it to one of you, and delegate him to buy what she needs.
According to some scholars, it is permissible to give her the zakaah from the outset in the form of food, clothing and other things that she needs. This view may be followed in cases such as this, if the son is going to take charge of the money and spend it on haraam things.
And Allaah knows best.