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, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is narrated on the authority of ‘Ataa’ ibn Yasaar that he said, “I asked Abu Ayyoob, ‘How was the Udh’hiyah done at the time of the Messenger of Allaah, sallallaahu ‘alayhi wa sallam?’ He said, ‘A man would offer a sheep on behalf of himself and the members of his household, and they would eat some and feed others with some, until people began boasting, and the situation now is as you see.’” [Ibn Maajah, At-Tirmithi: Saheeh]
This Hadeeth is an evident proof that people living in the same house can sacrifice one sheep or one seventh of a camel or a cow. It is narrated on the authority of Jaabir ibn ‘Abdullaah that he said, “We set out with the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, for Hajj. He commanded that we share camels and cows, and each seven of us share an animal.” [Muslim]
Thus, if that son and his mother live in the same house in such a way that they can truly be called members of the same household and they share the same source of provision, then one seventh of a cow will suffice them.
It should be noted that the cow must be two years old or more, because the Messenger, sallallaahu ‘alayhi wa sallam, said: “Do not slaughter but a Musinnah (a two year old cow or a five year old camel or a one year old sheep).” [Muslim]
Allaah Knows best.