All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
The basic principle is that it is impermissible to give your zakaah to your son. Ibn Qudaamah wrote, "It is not permissible to pay zakaah to one’s parents of all degrees or one's children of all degrees." [Al-Mughni]
Ibn Al-Munthir wrote, "The scholars unanimously agree that it is not permissible to give zakaah to one's parents in cases where the giver is legislatively obliged to spend on them, because giving zakaah to them means that they no longer have a need for his spending on them and makes him exempted from spending on them, and thus the benefit of that will come back to him; it is as if he is giving it to himself, and this is not permissible."
The scholars who held that it is obligatory upon you to provide for your son because he is unable to earn a living also held that it is impermissible to give him your zakaah, and this is the safer view. However, a group of scholars held that it is not obligatory on you to provide for your son because he is an adult; accordingly, you may give your zakaah to him. If you follow the view that it is allowed to give your zakaah to your son, then you must give him possession of the money when the lunar year elapses; it is impermissible to delay the payment of the zakaah under the pretext of saving the money for him.
It is more prudent to act upon the view of the scholars who forbid giving zakaah to the son under all circumstances. Accordingly, what you should do is urge your son to work and earn a living, to shun idleness, and to depend on himself in managing his affairs. If he needs money, you can help him, but not from the zakaah money.
Allah knows best.