For a basis, we're assuming modern day Greece, Cyprus, the western half of Turkey and additional lands in the Balkans, maybe southern Albania, Macedonia, and parts of Bulgaria as the surviving Byzantines.
The Byzantines will have the advantages of having a state that has survived and developed over the past 500 years. Population is probably better educated, and there is large elite class with experience in governance, military, and economics.
I also believe that this group fo Byzantines will be more open to western influence than the Ottomans were so western printing, armament, and other technologies will enter the empire and spread much earlier and quicker.
However, we don't know how moribund the empire will be. Long standing polities have entrenched interests that make reform harder. It will also have ongoing wars and raids by the Turks for a long period of time, forcing money to go into defense rather than economic investment and innovation. IOTL, the last few centuries of the Byzantines saw tremendous internal conflict between modernizers who wanted to reconcile with the West for survival, and a very spiritual, mystical Orthodox "nationalism" that was very anti-Western and saw the Latin world as polluting and alien (better the sultan's turban than the cardinal's hat as the saying went). A similar conflict will likely be ongoing in Byzantium during this time.
So I think a lot depends on specific events in this 800 year gap. All sorts of nations rose from nothing to become great powers and declined in this time period.
The most we can see is that a state that comprises Greece, Cyprus, western Turkey and some additional lands have a lot of potential. How well that becomes developed depends on how the internal Byzantine culture develops and how the record of bad and good rulers turns out in the end.
If we go with an average result between worst and best case scenarios, by 1900 we likely have a Byzantine Empire that is on par with or a bit better than Italy of the time period IOTL. It'd be a European great power that dominates the Eastern Mediterranean, but not a true world power like Britain, France, Germany, US, or Russia.