I think there would have to be an early POD for several reasons, mainly because if they want to conquer back of of their former land, they have to deal with native populations who have been thoroughly islamized and arabized. That means that the land is hard to control, and more trouble than it's worth. Even Egypt, though I know it pains people (byzantophiles) to hear that their beloved Rumania doesn't have the ability to conqueror, let alone hold, Egypt post-Muhammad.
For most of Byzantine existence, Anatolia was the border to the Muslim world. This meant the hallway for muslims looking to conquer into Eastern Europe, be it the Arabs, Persians, Seljuks, Turks (I know the Seljuks technically were Turks but they were a different wave than the Ottomans and their group), or whoever else. On top of that, the Mongols will very often be a problem, since they can come through the Caucus or Persia, and I imagine that they, like the turks, are a hard group to prevent from blobbing like in OTL. This means that the land (already mountainous) will be hard to work thanks to fairly common invading armies, and lack the infrastructure from the same said armies, will make industrialization hard, if they manage to get that far.
Another point is that, even if the Byzantines make it Renaissance in a better shape than OTL, they'll still have the problems that they had OTL, but on a larger scale, as well as typical eastern European problems. By the 1400s the byzantines were a system that was already struggling to keep up, and if they still control all of Greece and bits/all of Anatolia, this is going to be even harder to keep together, but with an early POD this can be relatively easily rectified. On top of that, They're in eastern europe, meaning that they'll be less developed because they have an overall lower population density, and don't have the centralization that the nation-states of the west have because they didn't have the hundred-years war or suffer from the plague near as bad, and neither of those are easy fixes, no matter the POD.
Outside of that, even if they manage to survive to the 1700s or so, the Byzies are in no position to participate in the ever globalizing world. Their position as the bridge between Europe and Asia is no longer an issue as going around is the norm now anyway, the blacks ea is pretty much useless to the westerners (who have all the important money) because its just got a bunch of russians, turks, and in this case greeks, living around it, and is out of the way to get goods they don't even really want/need. Because controlling Egypt is pretty much out of the question without the pre-700-ish POD, the Red Sea to the Far East suggestion is out of the question, especially since the Byzies wouldn't have been able to get a good fleet there anyway, because Egypt lacks any major ports on the sea I can think of at the time, so holding Arabia would be beneficial for the ports, specifically down the coat to Yemen, though that's even more impossible that Egypt, and if they did manage to get a large fleet in the Red Sea, they then have to get a sizeable amount of troops to it (large enough to conquer land from either the Indians or Chinese, who have large populations, even then), which is hard considering its all desert around there. To top it off, a hostile navy from the non-Byzie controlled Yemen could pretty easily destroy a fleet in the Red Sea, since it's narrow and has a chokepoint exit. Going through the straits of Gibraltar is unlikely, since they would first have to conquer it from the west, a task improbable in and of itself, as well as prevent someone stronger still from taking it again. If they don't control Gibraltar, then whoever does control it can just decide it doesn't want Greeks in the atlantic, and blow their fleet to Timbuktu, though that might be what the greeks want...
So to summarize, I think any surviving Byzantine Empire, Rumania, Rhomaion, whatever, would most likely be a slightly larger greece with a slightly larger economy to match.