In which case it would be a hell of a coincidence, since romii also directly connects to the Eastern Roman Empire (aka the Byzantine Empire):
"The national name of our ancestors throughout all these years is “Romans”, or “Romee” (in Greek “Romií”, Pronounced “Rome-ee-ee” Plural for “Romeos”), in the popular form of the language. In every one of the historical sources, without exception, the Empire of Constantinople refers to itself as “Roman”, or “Romania” (=land of the Romans) in the popular form of the language, while its emperors, up to and including Constantine Paleologos, were known as “king of the Romans”."
source (chapter 2 - So, Hellenes or Romans?).
That's pretty arcane, but I concede the
possibility that it might be related.
edited to add: It would seem that there are different schools of thought on how to render the term in the Latin alphabet.
One of the issues I have with the rendition "Romii" is that various sources, including yours, claim that this is a Greek term, and yet it is not rendered in the Greek alphabet. The actual Greek term in question appears to have been Ρωμαίοι or Ρωμηιοί. The sources cited in the Wikipedia article quoted below render the term
in Latin in the Latin alphabet as "Romioi," which is closer to a straight transliteration.
Plus, this is the name of a people, not a place.
From
https://en.wikipedia.org/wiki/Ottoman_Greeks#Identity:
The Greeks were a self-conscious group within the larger Christian Orthodox religious community established by the Ottoman Empire.[7] They distinguished themselves from their Orthodox co-religionists by retaining their Greek culture, customs, language, and tradition of education.[7][8] Throughout the post-Byzantine and Ottoman periods, Greeks, as members of the Ecumenical Patriarchate of Constantinople, declared themselves as Graikoi (Greek: Γραικοί, "Greeks") and Romaioi or Romioi (Greek: Ρωμαίοι/Ρωμηιοί, "Romans").[9][10][11]
[...]
7. Harrison 2002, pp. 276–277: "The Greeks belonged to the community of the Orthodox subjects of the Sultan. But within that larger unity they formed a self-conscious group marked off from their fellow Orthodox by language and culture and by a tradition of education never entirely interrupted, which maintained their Greek identity."
8. Volkan & Itzkowitz 1994, p. 85: "While living as a millet under the Ottoman Empire they retained their own religion, customs, and language, and the 'Greeks became the most important non-Turkish element in the Ottoman Empire'."
9. Kakavas 2002, p. 29: "All the peoples belonging to the flock of the Ecumenical Patriarchate declared themselves Graikoi (Greeks) or Romaioi (Romans - Rums)."
10. Institute for Neohellenic Research 2005, p. 8: "The people we have named as Greeks (Hellenes in the Greek language) would not describe themselves as such – they are generally known as Romioi and Graikoi – but according to their context the meaning of these words broadens to include or exclude population groups of another language and, at the same time, ethnicity."
11. Hopf 1873, "Epistola Theodori Zygomalae", p. 236: "...ησάν ποτε κύριοι Αθηνών, και ενωτίζοντο, ότι η νέων Ρωμαίων είτε Γραικών βασιλεία ασθενείν άρχεται..."
