I have noticed that virtually all Roman emperors after Constantine were called Flavius Something. A quick lookup in wikipedia confirms this and even more:
During the later period of the Empire, the name Flavius frequently descended from one emperor to another, beginning with Constantius, the father of Constantine the Great. The name became so ubiquitous that it was sometimes treated as a praenomen, to the extent of being regularly abbreviated Fl., and it is even described as a praenomen in some sources, although it was never truly used as a personal name. The last emperor to take the name was eastern emperor Constantine IV.
After the name fell into disuse among the Byzantine emperors, it was used as a title of legitimacy among the barbarian rulers of former Roman provinces, such as Spain, where the Visigoths and their Spanish successors used the title ″Emperor of All Spain″, and the kings of the barbarian successor kingdoms of Italy, such as the Ostrogoths and the Lombards also used it, with a special meaning as the ″protector″ of the Italian peoples under Lombard rule.
So, I looked up Chlorus and apparently he started calling himself Flavius when he was appointed Caesar in 293 (wiki again):
At Milan on March 1, 293, Constantius was formally appointed as Maximian’s Caesar. He adopted the names Flavius Valerius and was given command of Gaul, Britannia and possibly Hispania.
But why did he choose specifically Flavius?
A cursory search reveals that in the 3rd century a popular soi-disant name for emperors was Aurelius and that Diocletian called himself Aurelius Valerius (so that explains the Valerius part all right).
Was Constantius trying to be special? Was he claiming some particular affinity to the real Flavian dynasty of the 1st century?
What makes this a bit more weird is that apparently Constantius did have a fabricated genealogy but there was nothing Flavian about it:
Born in Dardania, Constantius was the son of Eutropius, whom the Historia Augusta claimed to be a nobleman from northern Dardania, in the province of Moesia Superior, and Claudia, a niece of the emperors Claudius II and Quintillus. Modern historians suspect this maternal connection to be a genealogical fabrication created by his son Constantine I, and that his family were of humble origins.
But Claudius II's full name was Marcus Aurelius Valerius Claudius Augustus and his brother Quintillus was Marcus Aurelius Claudius Quintillus Augustus (note the weird absence of Valerius). No Flavius there! Of course, if the whole thing was invented after Constantius's own time, retconning it properly might have been too complicated.
Maybe Constantius just liked the sound of the name Flavius?