We as a society identify people by their last name (or sadly, even if understandably, by a number). Such was not always the case. When life was simpler and populations smaller people were on a first name basis something that today would make it impossible to locate people. Perhaps reflecting the small Jewish world in which we live and expressing a desire for closeness that a first name indicates, our tradition still refers to people by their first name. Demonstrating respect for our parents we add their name to ours. This is likely a fulfillment of the rabbinic dictum that one should attribute all honour one receives to their mother or father.
This discussion is not merely semantics but has legal ramifications. "Ploni eid, so and so, witness it is valid." (Gittin 87b) So teaches the Mishna as it validates a get even if the witnesses signing it use their first names only. One can even sign the get using one's father's name as in "the son of so, a witness it is valid". The Mishna then lists a third option "Ploni the son of Ploni but he did not write 'eid, witness' it is valid".
The phrase "it is valid" implies that ideally one must include the phrase eid but that b'deavad, if one did not include it the get is still valid. And such is the practice today that when signing a ketuba or get we add the descriptive phrase eid, after signing our names.
It is thus somewhat surprising to read the next line in the Mishna "and such did the nekeyei Yersushalayim", the clean ones, i.e. the rabbinic elite of Jerusalem, writing only their name and the name of their father. Why this is so is not explained and seems quite out of character. We have met the nekeyei Yersushalayim before, for example in Masechet Sukkah where they would carry their lulav and etrog with them wherever and whenever they went on Sukkot. As people who were most meticulous in their mitzva observance it seems strange that they would not be careful to add eid after they sign their name.
Rashi explains that these pious people did not want to waste "words" thus they signed in brevity. This is a rather astounding theory and while I am not sure it fully explains the custom it is a powerful message in the importance of being careful with every word we utter or put in print.
Tosafot however offers a different explanation, one that highlights another mitzva. The Mishna continues that if instead of writing one's given name one wrote chanichato the get is valid. It was here Tosafot claims that the line "this was what the nekeyei Yerushalayim used to do" is to be inserted.
Rashi and Tosafot explain the chanichato is a family name. Thus according to the Tosafot the pious ones of Jerusalem used not their given names but their family names when writing a get . So while the Mishna is ostensibly discussing the halachic minutiae of writing one's name in a legal document we have a beautiful dispute as to whether the mark of piety is brevity of words or displaying honour to one's family. As in so many rabbinic disputes there is no need to reach a definitive conclusion; "these and those are the words of the living G-d."
 Presumably it would be no different if one signed as the son of one's mother. Our custom of identifying ourselves through our parents is more a convention than law. As we all know when being called to the Torah we use our father's name as (traditionally) only men are called to the Torah, but when praying for a speedy recovery for the sick we use the mother's name as it is the woman who is the symbol of mercy. It is not unheard of today, even in Orthodox circles to use both parents' name reflecting the changing status of women.
 The language of the Mishna and as codified in the Shulchan Aruch implies that only when one used both his name and his father's name is the get valid if the term eid is missing. In the case when only one name is used the term eid becomes obligatory. I do not fully comprehend why this would be so.
 While family names did not come into vogue until the Late Middle Ages at least according to this explanation of Rashi, Jews of the ancient world had some form of family name. Rashi does offer another explanation that chanichato refers to a nickname.
 While one might argue that putting a family name in a divorce document brings little honour to the family name the laws of signing a get apply to all legal documents. The pious ones of Jerusalem subsumed their names to that of their family seeking to highlight others rather than themselves.