While reading on Dioscorus of Aphrodito, I noticed that while the poet's complete name was “Flavius Dioscorus,” his father was simply called “Aurelius Apollos.” The book I was skimming through remarked that it was due to social promotion, because the name Flavius denoted a higher social rank; and indeed, important people of the Late Roman Empire tended to be called Flavius. Since Diocletian complete nomen was “Gaius Aurelius Valerius Diocletianus Augustus”, I assume that the transformation of Flavius as a sign of social standing postdates the end of the 3rd century. When did this change happen? Why?
It has to do with the meaning of the name itself - yellow, golden, blond. Here, I believe, it refers more to complexion than to wealth, though that is certainly an aspect.
I am wracking my brain for where I've read this - I don't think it was Peter Brown, but I don't read much other than him for 4-5th c. - but it seems to have been one of those bits of "barbarian" culture which were adopted in this period along with pants. It would have thus carried connotations of skill in battle, etc.
When I remember the source, which will probably be tomorrow afternoon in the shower or something, I will provide it.
I did a bit of digging on this as I found it rather interesting, and found a couple articles of note on the phenomenon of the use of Flavius and Aurelius in the Late Empire. My first thought was that the two names were simply titles taken from previous emperors (Titus Flavius Vespasianus, Marcus Aurelius) for government offices, but there seems to be a bit more involved.
J.G. Keenan has looked at the names Flavius and Aurelius in Late Roman Egypt, and has found that, generally, the Flavii were the wealthy landowning classes, designating Imperial civic officials or former Roman soldiers now settled in the province and that the Aurelii were 'Egyptian' farmers and craftsmen who would lease from the Flavii, or members of local councils (curia) and well-to-do villagers. This is grossly simplified, as these are most definitely not hard and fast categories, but merely patterns that Keenan found in his rather methodical work (looking at papyri records of leases and fasti).
That would then assume that the "Aureliate" was of a lower social standing than the "Flaviate" in the late antique period; this would explain the generational difference in cognomen between Flavius Dioscorus and his father.
However, A. Cameron of Columbia stresses that this is a strictly Eastern phenomenon and that the West utilized the name "Flavius" very differently. Cameron believes that the name Flavius was just a "nicety of protocol," equivalent in a very general sense to the English 'Mr.' I'm not sure I agree, but I do find the explanation rather interesting.
But from what I'm reading, these particular authors are using 4th-5th century CE records, so your hunch that it postdates the 3rd century seems pretty spot-on. However, neither Keenan nor Cameron give an explicit date or event that caused the change which leads me to believe that there might not be one. I'm sure there is more scholarship on the issue as both authors are quite dated, but if you have access to JSTOR you can look at the following.
Keenan, J.G. 1974. "The Names Flavius and Aurelius as Status Designations in Later Roman Egypt."
Cameron, A. 1988. "Flavius : a Nicety of Protocol."