The rise of 'Flavius' in the period you are interested in is mostly due to the Constantinian dynasty, which held power between Constantine I and Julian (306-363): Flavius was their family name (gentilicium). This name became widely adopted by the elite, as it denoted status and generally indicated that the person served the imperial government, whether at the centre or in the provinces. Those of high social status were able to use 'Flavius' instead of the more common 'Aurelius' once they reached a certain rank. It wasn't always hereditary, as their sons sometimes took the name "Aurelius" if their careers changed (a name that indicates a lower status, but in fact the most attested to name in late antiquity since there were more of them than the elite Flavii). Another example would be the name Valerius, which became very popular before Constantine I, as it was the family name of Diocletian, the man who dominated the Tetrarchy.

Basically, in the period you are looking at Flavius had become more of a title than anything else (an analogy might be 'Mr' in modern English) and adopted by those involved in the administration, which lasted beyond the Constantinian dynasty itself. 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).