All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His Slave and Messenger.
The first: If they are in debt and unable to pay the debt. In this case, it is allowable for her to pay her Zakah to her child to pay off the debt because she is not obliged to pay off the debts of her poor children. She is obliged to provide for her poor children only (and not pay off their debts).
The second: Some scholars held that if she is financially unable to provide for her poor children, then she may pay her Zakah to them. This is because it is only impermissible for her to pay her Zakah to her children when she is obliged to provide for them. However, if she is financially unable to provide for her poor children, then she may pay her Zakah to them. This is the opinion of the Shaafi‘i scholars and the preferred opinion of Shaykhul-Islam Ibn Taymiyyah and Ibn ‘Uthaymeen and this opinion is perhaps the most evident one.
To sum up, if your son is poor and you are financially able to provide for him, then it is impermissible for you to pay your Zakatul-Fitr to him. Instead, you are obliged to provide for him and pay his Zakatul-Fitr as well. But if you are financially unable to provide for him, then it is allowable for you to pay Zakatul-Fitr to him as long as he is an eligible recipient.
For more benefit, please refer to Fatwa 86579.
Allaah Knows best.