At some point in the late third century the praenomen Flavius - if indeed that's what it is - seems to become extraordinarily prevalent among prominent figures in the Roman Empire.
Not only does almost every Emperor from Constantine the great all the way to Constantine VI some 350 years later carry the name, but seemingly unrelated people. Roman generals such as Aetius, Aegydius or Belisarius as well as romanised germans like Stilicho, Abrogast, Ricimer, Odoacer etc are also either christened or late take the name Flavius. Why?
'Flavius' is not a praenomen, but instead a nomen, which is more of a surname than a first name. Already in the Principate period the praenomen was not used to distinguish between different members of the same family - many family members for example all had the same praenomen, best seen in the names of the emperors Vespasian, Titus and Domitian, who all had the praenomen Titus. What differentiated them instead was their cognomen. Later under Caracalla citizenship was granted to all free men living within the empire and so an overwhelming number of people took on 'Aurelius', the name of their benefactor, as their new Roman nomen. Basically it had more to do with a client-patron relationship rather than it being an indicator of patrilineal descent. The same process in a different context was the norm for slaves freed by their masters for example. From then on 'Aurelius' would be the most commonly attested to name in Roman history, simply because most people possessed it, and should be seen more as a status symbol rather than a personal name. It might be better to think of it as a title like 'Mr' rather than a name per se.
But why were there so many 'Flavii' instead? Well, both 'Aurelius' and 'Flavius' denoted status and after Constantine I 'Flavius', the nomen of his dynasty, became associated with the imperial government. Those of high social status were able to use 'Flavius' instead of the more common 'Aurelius' once they reached a certain rank. As I said, the nomen at this point was no longer an indicator of familial ties, but now symbolised someone's social position. This is why practically everyone in positions of power adopted the name 'Flavius' as well - they were after all the elite of the empire and close to the emperor, it would be more strange if they did not have that name! The same was true even after the 'fall' of the western Roman empire, as its 'barbarian' successors were no less keen than 'real' emperors to trumpet their connections with what the empire represented.
Further reading:
J. Keenan, 'The names Flavius and Aurelius as status designations in later Roman Egypt', Zeitschrift für Papyrologie und Epigraphik (1973 and 1974).
B. Salway, 'What's in a Name? A Survey of Roman Onomastic Practice from c. 700 B.C. to A.D. 700', The Journal of Roman Studies (1994).