1. Da Qin “Rome; Mediterranean Civilization” 大秦
Middle Chinese / d͡zʱi̯ĕn/ — possibly connected, via Christian missionaries or Jewish trading networks, to “Zion” (TsYN/Sehyon/Tsiyyon)
One reconstruction of 大 in Old Chinese gives /lˤat-s/; but “Latin, Latium” /lˤat(s) d͡zʱi̯ĕn/ seems unlikely, both because the Silk Road was opened too late, and because (so far as I know) people in the Eastern Empire tended to refer to the political/cultural sphere of the Empire as Roman and use Latin as a toponym, etc.
The mysterious cult of the Therapeutae have been interpreted as missionaries from the Theravāda tradition of Buddhism. I have not been able to find any evidence that the term Theravāda was used to denote the conservative varieties of Buddhism (i.e., non-Mahayana) until centuries later. However, Tripitaka as a description of the doctrine or scriptures of some Buddhist sect seems like a much more plausible source of the term; the initial Thera- could have arisen from native Copts difficulties with initial tri-; from differences among the monks about the proper pronunciation; or from a conflation of tri-/ti- with the title of respect for an elder, Thera.
The Hayaza/Azzi kingdom or tribal confederation operated circa 1400-1200 BC, and came into conflict with the Hittite (Neša) Empire. Many others have proposed some connection between the components of this name and the Armenian endonym “Hay” (cf. legendary patriarch “Hayk”) or exonym ’Arminoi.
More specifically, I suggest “Hayasa” derives from PIE *ekw-, “horse”, which has reflexes like yäkwe, áśva, hippos; something like *hyas may be possible for some branch, for which “Ha-ya-sa” would be the Hittite transliteration. (Sanskrit in fact has háya “horse”, but this is from a different root *ǵʰey- and so would have had to be borrowed directly from an Indic language, like that spoken by the rulers of the Mitanni until circa 1300 BC.)
“Azzi” on the other hand, could come from a PIE root meaning “to flow” yielding the connotation “male, men”, with reflexes like Greeek ’arrenikos, ’arsenikos (cf. ’ersê “dew”), Sanskrit vrsan “cf. rs- “flow”, vrs- “to rain”), Avestan(?) arśan, Germanic wair, wer, etc. Given the complexities of rhotics, it is easy to imagine the Hittites transcribing some dialectical variant as “Azzi”.
If these guesses are correct, then Hayasa-Azzi means something like “horsemen”, which would be a very standard endonym; it could also be the endonyms of two separate groups. Additionally, it could be the endonym(s) of one group, calqued into the language of another group. For example, if the group called itself something like *Bhrygi-mushki (cf. Proto-Celtic *uɸorēdos “horse” and Old Slavic muški “man”) in one of the lost IE languages and then came under the leadership of the remnants of the Mitanni military hierarchy, the latter might have translated “men of the horse” as Hayasa-Azzi.