In many medieval films we see men introduce themselves this way. For example, "I am Edward, son of Gregory." If this was common, was it by most men or just nobility?
I'm not sure if men physically introduced themselves as such but I do know that this was one of the forms with which men were known. There were multiple ways with which men were known and each of these, over time, evolved into the surnames we have today. Off the top of my head from the books I've read I know there are three styles of surname origin:
A quick search on wikipedia shows that I forgot two additional categories in nicknames/descriptors, and ornamental names.
In the first category, given names, you have both your example where names like Williamson and Johnson derived from William's son or son of John, as well as clan names. In addition to patronymics (being named after your father) you also have matronymics (being named after your mother).
Location names came about in two ways. The first was with royalty. The name would link them with their place of origin and give an idea of their standing in society. The second path to derivation was with the rise of of respectably sized towns and cities. As you bring in more people you have to have ways to differentiate them. The easiest thing to note was if men originated from other communities. Daniel Radcliff, Jack London, and Edward Norton all have location based names.
The third category is occupational based names. Many common names you can think of fall under this category: Miller, Smith, Booker, Barber, Baker, Barker, Carter, Carver, Tanner, Hunter, and the list goes on.
I'll include nicknames/descriptive names because I thought of a few as I wrote this response (including my own name). Color names like Brown, Black, Gray, Green, Blue, and White usually referred to someones hair color, eye color, or complexion. Other names like Short came from physical descriptions not related to color. You could also have literal nicknames that over time became the family name. My last name falls into one of the later to subcategories. It's either a pet name for someone named Erasmus or a descriptive name for someone who looks like a fish.
So surnames primarily derived in larger cities as more people came together and you needed to differentiate between individuals. You needed to know the difference between Jack, The one with black hair, and Jack, the one from London. The custom spread as migration became more common and eventually certain names stuck. You went from being called John's son because your father was John to William's son because your father was also know as William's son even though you were the son of John. Sorry I don't have the books on hand to say whether they would have introduced themselves that way but it was clear that men would have been known that way.
Sources:
In Iceland we still do this, in a way. Our last names are the name of our father, postfixed with "son/daughter". E.g my last name is "Hannesson", or son of Hannes
I used to work with the elderly in Holland. (Specifically the provence of Frisia, with the Frisian language and all) and I noticed alot of the elderly would reference the father of the person they were talking about.
For example: Jaap, fan Timen hat in neie auto kocht.
Translation: Jaap of Timen has bought a new car.
Translation with common names: Jason, of Bob has bought a new car.
Just my two cents.
It was very common in the islamic world to even refer to prior generations than just the father. To take the most known muslim man in history the prophet mohamed's actual name was Abū l-Qāsim Muhammad ibn ʿAbd Allāh ibn ʿAbd al-Muttalib ibn Hāschim ibn ʿAbd Manāf al-Quraschī ibn means "son of".
Even people whose father was not known were named ibn abihi which means son of his father.
Source: Coran 33:5 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.
Do songs count as a good enough source here? Dating from at least the 17th Century, the English folk song, "Little Musgrave" has the son of a landowner being addressed in that way throughout.
It appears in Wit and Drollery (1658). A copy was also printed on a broadside by Henry Gosson, printed between 1607 and 1641. Circa 1611 it is quoted in a play titled Knight of the Burning Pestle under its alternate title, "Mattie Groves".
This is my favourite version of the song, by Martin Simpson.
A lot of this discussion seems to concern English practices. But, what about in Italy, for example. During the Roman Republic, there was a very robust system of praenomen, nomen, and cognomen. Did Medieval Italians preserve this system (likewise, how much of it exists to this day)? If not, why did it die out?
It is a way that some would identify themselves. An example is the Mc/Mac prefix used in Scottish and Irish surnames is Gaelic for "son of".
What about Eastern European naming styles of the same time period?