Hmm? The people during the Prophet's (saw) time didn't keep any 'family names'. It was usually ____ bin ____ or ____ bint ______.
So (the name of the child) bin/bint (the first name of the father). Isn't that how it was?
This is not just the prophet (PBUH) time, this is still practiced today as i do. The western system however is different and you follow you father by his surname, unlike us who are like fulan bin/bint fulan.
It is reported in a Hadith that the Prophet -peace be upon him- said, "You will be called on the Day of Resurrection by your names and the names of your fathers, so have good names." (Sunan Abu Da'ud, Hadith 4297)
During the time of the Prophet -peace be upon him- when people used to accept Islam, he would not require all people to change their names. He used to change only those names that carried wrong meanings or un-Islamic ideas and concepts. The Prophet's wife 'A'ishah -may Allah be pleased with her- said about the Prophet, "He used to change bad names." (Sunan al-Tirmidhi, Hadith no. 2765)
'Umar had a daughter whose name was 'Asiyah (meaning Tough or Hard). The Prophet changed her name to Jamilah (the Beautiful, see Sunan Abu Da'ud, Hadith no. 4301)
Abu Da'ud has mentioned many names of people and places that the Prophet -peace be upon him- changed because they had bad meanings. (see Hadith no. 4305)
Imam Bukhari mentioned that Sa'id bin al-Musayyib said that his grandfather came to the Prophet -peace be upon him. The Prophet asked him his name. He said, "Hazn" (meaning Hard). The Prophet said, "You are Sahl." (meaning Soft, Easy). The man said, "I do not want to change the name that my father gave me." His grandson Sa'id used to regret and say afterwards that we still experience the hardness in our family. (see al-Bukhari, Hadith no. 5722).
One can only change one's own name. One cannot change the family name or name of one's parents, unless they are alive and accept the change. So the new Muslims should change their first names, if they contain wrong or bad meanings. As far as their last names are concerned they should not change last names, because it is also forbidden in Islam to take the name of someone other than one's own biological father. Allah says in the Qur'an, "Proclaim their real parentage. That will be more equitable in the sight of Allah. And if you do not know their fathers, then they are your brothers in the faith…" (Al-Ahzab 33:5)
Allah know best