Must a Child be Given his Father's Name?
Is there any ayat or any hadith or any hadith saying that in this world, children will be called by their fathers' names?
To give the child the name of his/her father is a social and not a religious tradition. The directive of the Quran, given in a somewhat related issue is that a child should not be ascribed to some one who is not his real father. The context of this directive relates particularly to adopted children (see: Al-Ahzab 33: 4 - 5). In essence, the Quran has directed us that in case of an adopted child, to avoid all the social problems that may arise in future, he/she should be ascribed to his own real father and not to the adopted father or anyone else.
Besides this verse in the Quran, which pertains only to adopted children and the social problems that may arise in future if they are ascribed to the adopted father, there is no specific directive that a child should be given its fathers name.
On the other hand, according to the social tradition of the Arabs, the child was given the name of his father. Obviously, if there was something wrong with this tradition, in the law of Allah, it would have been corrected during the time of the Prophet (pbuh). The Quran, as you know has not only accepted this social tradition but in the case of adopted children has stressed that they should only be ascribed to their real fathers. This, in my opinion, shows that there is nothing wrong in following this tradition of giving a child the name of his/her father.