Is the name of "Diocletian" just a coincidence?

Diocletian's original given name was Διοκληϛ (Diocles), which is a very common Greek first name.

The meaning of the name is uncertain, but διοικεω means "to rule" and the first part of the word (DIO) has the sense of a god or a lord.

Your guess is not completely off base, however, because words for "two" are similar. For example, the word διακλαω means to break something into two pieces. If this is the case, it would have to be very old because dio- in the sense of two would only have been used archaically.

Another possibility is that the word Διοκληϛ might have been given in very ancient times to a second child, just as the Romans used to sometimes call their second child "Secundus".

Comment on Δίο meaning Zeus or of Zeus

There is a persistent thread of thought dating back even to ancient times that Δίο somehow refers to Zeus in particular. This idea appears to have originated with classical era playwrights such as Aeschylus and Euripides, who appear to have used Δίο to mean Zeus as sort of a literary conceit. However, to take this usage from Aeschylus and infer that Dio means Zeus is absolutely without basis, because in all older sources διο unquestionably simply means divine or godlike. For example, in Homer the word is used simply to mean divine. For example, in Book 9 verse 538 of the Illiad it says "δῖον γένος ἰοχέαιρα" meaning divine (see Perseus: http://www.perseus.tufts.edu/hopper/text?doc=Perseus:abo:tlg,0012,001:9:538). There is even the feminine form δια meaning a divine female, not to mention the Latin word deus which dates to before 600 BC and means just "a god", not Zeus.

It just a coincidence. I think what's misleading here is the similarity between those Greek words:

1. Διος (Dios) meaning "of ZEUS"
2. Δύο (Dio) meaning the number two

Also, there's the fact that Diocletian did split the empire in two sections.

I'll try with an answer as well, trying to be Solomonic between @Tyler Durden and @fdb.

TL;DR: Yes, it's a coincidence.

As @Peter Diehr lists from Behind the Name entry for Diokles the meaning is given as

Given Name DIOKLES

GENDER: Masculine

USAGE: Ancient Greek

OTHER SCRIPTS: Διοκλης (Ancient Greek)

Meaning & History Means "glory of Zeus" from Greek Διος (Dios) meaning "of ZEUS" and κλεος (kleos) meaning "glory".

Διος is the genitive.

The etymology of Zeus is given in the Online Etymology Dictionary as

supreme god of the ancient Greeks and master of the others, 1706, from Greek, from PIE *dewos- "god" (cognates: Latin deus "god," Old Persian daiva- "demon, evil god," Old Church Slavonic deivai, Sanskrit deva-), from root *dyeu- "to gleam, to shine;" also the root of words for "sky" and "day" (see diurnal). The god-sense is originally "shining," but "whether as originally sun-god or as lightener" is not now clear.

κλεος makes an appearance in a number of other words, such as the latinised Cleon, and also the name of the muse Clio

[the] muse of history, muse who sings of glorious actions

and also, Hercules, in its Greek form Heracles, itself derived from Hera (wife of Zeus) and kleon, giving us Glory of Hera.

Wikipedia gives the -anus ending as adoptive cognomen, meaning a suffix to indicate Diocletian would have been adopted (not unusual in the Empire). For those allergic to citations from Wikipedia, here also Roman Nomenclature

Adoption: An adult son of a family which already had a male heir could be adopted into a family which did not have a surviving son. The adopted man took all three names of his adoptive father and usually added the adjectival form of his own clan name, formed by adding the suffix -anus) to his own nomen. Thus, when Gaius Octavius Thurinus was adopted by his great-uncle, Julius Caesar, his formal name became Gaius Julius Caesar Octavianus. At the time, those who addressed or referred to him respectfully would do so as "Caesar" or "Gaius Caesar"; those who wished to be denigrating or disrespectful would use his adoptive, "Octavianus." Modern historians usually call him Octavian until he officially added the honorific Augustus (“the revered one”) to his name in 27 BCE

According to De Imperatoribus Romanis Diocletians full name was

Gaius Aurelius Valerius Diocletianus

It is a Latin derivative of the Greek name Dioklēs, which is from dio- (the compositional stem of the divine name Zeus), plus –klēs (“fame”). So it means “with fame from Zeus”. It has nothing to do with di- “two”.

Reference: Beekes, Etymological dictionary of Greek.