This is a really vague question, so there are a million PODs, from Yarmouk to Mantzikert to the Fourth Crusade. Hell, until it took back Byzantium did Nikaea stand a chance.
If you want one such thread of surviving Byzantium, check out "An Age of Miracles : the Revival of Rhomanion" by Basileus444.
A Yarmouk POD means no Islam in Iberia and North Africa and possibly a recreation of the Roman Empire.
A Mantzikert POD means no Islam in a Balkans, which means much more infighting in Central Europe, and the end of the Sultanate of Rum. Basically, I see surviving Crusader states.
A Fourth Crusade POD leaves Byzantium a chance to have a lucky ruler who reconquers Anatolia. Still, probably more time before the fall of Byzantium.
A Nikaea POD... Well it's An Age of Miracles.
With a Manzikert POD the Byzantine Empire would still be in control of Asia Minor and this area would still be non-muslim, so you could very well have a much later POD to keep a Muslim power far away from Balkan.
Personally I think you can achieve this even after the Ottomans take Gallipoli, the PoD just has to get incredibly large.
Personally, I'd be interested in a No Osman, or an Osman of Constantinople PoD. If he is captured at a young age (say his father is defeated in a battle), and converted - we could see him lay claim to the leadership of his fathers Turkish troops, which wouldn't be easy, but not ASB, and either as a vassal, General, or appointed heir (don't ask me how, I'm spitballing), he could bring in a large Turkish domain into Byzantine hands, reinvigorating the Romans.
Alternatively, just have a freak accident in the Osman household. Another Turk comes into play who isn't as talented, and perhaps destablises the Turks enough to give the Byzantines an opportunity that is taken.
This is a really vague question, so there are a million PODs, from Yarmouk to Mantzikert to the Fourth Crusade. Hell, until it took back Byzantium did Nikaea stand a chance.
If you want one such thread of surviving Byzantium, check out "An Age of Miracles : the Revival of Rhomanion" by Basileus444.
A Yarmouk POD means no Islam in Iberia and North Africa and possibly a recreation of the Roman Empire.
A Mantzikert POD means no Islam in a Balkans, which means much more infighting in Central Europe, and the end of the Sultanate of Rum. Basically, I see surviving Crusader states.
A Fourth Crusade POD leaves Byzantium a chance to have a lucky ruler who reconquers Anatolia. Still, probably more time before the fall of Byzantium.
A Nikaea POD... Well it's An Age of Miracles.