Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
The mother should not give the Zakah of her money to her son because she should spend on him if he cannot spend on himself. So, if she gives him her Zakah it would be like protecting her money. But if the son is indebted, then there is no objection to her giving him from her Zakah what could pay his debts. Ibn Qudamah said: 'It is compulsory on the child to spend on his mother as it is compulsory on the woman to spend on her child if he has no father (dead or the like). This is the opinion of Abu Hanifa and Al-Shafie . It is reported from Malik that she is not asked to spend on him because the expenditure is due on the males from the father's side…... But since she is one of the parents she is then considered like the father and because the relation between them (the woman and her child) is so important that the testimony of one of them for the other is rejected in Shariah. Now if the father is poor so, the mother has to spend on the children' .
Allah knows best.