The question is somewhat problematic as I don't quite understand what sequence of PODs is being proposed. The strategy for a seventh century Empire would differ from the eleventh which would dramatically differ from the thirteenth and so on. I'll thus give a hyper general answer:
Honestly, all they'd have to do is survive. Then modern communications and a Greek education system would do the rest. Let's not forget how many different languages there were in OTL Italy/France in the 18th/19th Centuries and what happened to them since then. Most of the constituent people within the Byzantine Empire would be Hellenized when the time for it came-we were dealing with a highly literate society with centralized administration. Ambitious young men could advance their career through the civil service, which would mean Hellenization. Additionally, they have a long history of population transfers (OTL Greece was de-slavisized in that manner) and so a particularly independent minded minority could find themselves on the opposite side of the Empire.
The Turkish comment is interesting-Turkic people in Anatolia pre-Manzikert were quite Hellenized already. If the Byzantines lost Manzikert so that loads of muslim Turks swarmed in, but later pulled off a reconquest, I would suspect that the Turks who would remain would hellenize rapidly (the same way Greeks Turkified). It was in the interest of the conquered people to speak the language of their conquerors in that part of the world (see the Arabisization of Levantine Christians and Copts, as well as the extreme case of the Karamanilides-
https://en.wikipedia.org/wiki/Karamanlides). Plus, identity is tied too strongly with religion, and success in converting Turks to Orthodoxy would likely hellenize them substantially in the process. So long term, all they have to do is survive and be in better shape than Ottomans or Austro-Hungary when the moment of reckoning came (honestly though, most reasonable PODs would have the Byzantines holding their Empire for ~800 years ish continuously, and so they may not even have a minority issue worth thinking about by then.) If we assume static 1025 borders for instance, they'd have an almost purely Greek Empire (I think the Armenians would have probably endured-judging from their history) by today.