The Quran itself in Surah Al-Azhab tells us to be called by our fathers name for example Ibn Kathir which means 'son of Kathir'.
Allah says in the Qu'ran:
Call them by [the names of] their fathers; it is more just in the sight of Allah . But if you do not know their fathers - then they are [still] your brothers in religion and those entrusted to you. And there is no blame upon you for that in which you have erred but [only for] what your hearts intended. And ever is Allah Forgiving and Merciful.
This verse clearly states the importance of being named after our fathers. Instead of being called our father's name, people include our father's name in our name hence why people are called Ibn meaning 'son of' or can be called Bint ( if they are a female) which means 'daughter of'. For example, instead of calling 'Ibn Kathir', 'Kathir', you would say 'Ibn Kathir'.
It was also in the Pre-Islamic Arab culture to name one after their father and their father etc.