The child is ascribed to the father, not the mother: his/her name should be followed by the name of the father.

Just as the naming of the newborn is the right of the father, likewise the newborn is also ascribed to the lineage of the father, not the mother – and he is addressed with his father’s name, not his mother’s name. So, he is named as follows: “So-and-so the son of… father’s first name” or, “so-and-so the daughter of… father’s first name.”

So if, for example, the father names his child Abdullah, and the father’s name is Abdul-Kareem Hasan, the child’s name would be Abdullah Abdul-Kareem Hasan.

So, when the child is called upon or addressed, it is said: “O son of so-and-so (father’s name)” or “O daughter of so-and-so (father’s name).” Allah (the Most High) said:

ٱدْعُوهُمْ لِـَٔابَآئِهِمْ هُوَ أَقْسَطُ عِندَ ٱللَّهِ

“Call them (children) by the names of their fathers: that is more just with Allah.” (Al-Ahzāb: 5)

Imam Al-Bukhari has a chapter heading in his saheeh:

باب مَا يُدْعَى النَّاسُ بِآبَائِهِمْ

“Chapter: Calling the people by their father’s name.” Then he reports the hadeeth where the Prophet (salallāhu ʿalaihi wasallam) said: “For every betrayer, a flag will be raised on the Day of Resurrection, and it will be announced: ‘This is the betrayal of so-and-so, the son of so-and-so (i.e., the father’s name).’” (no. 6177)

(See: Tasmiyat al-Mawlood, p. 29-31, of Shaikh Bakr Abu Zayd)

Read also: Rights of Children Born of Fornication and Ascription to their Fathers ― by Shaikh Ibn Bāz