Well, a few ideas:
1) Avoid Manzikert entirely and keep Anatolia
2) Avoid 4th crusade or buy them off and redirect them somewhere else
3) Have Ottomans claim Byzantine throne through marriage with rump Ottoman state remaining after avoiding WWI
4) Permit Morea or Trebizond to exist as minor satellites of the Ottomans, slowly expanding influence until Morea controls much of Greece circa 1885 and Trebizond controls much of northern Anatolia. Let them go independent but separate
5) Get Greeks and Serbs to work more closely together in late 19th century and divide Balkans among them, especially at the expense of Bulgaria
Maybe something like this...?
Success at Varna shortly before 1453 permits Byzantines to exist as a smaller trading state but does grant them some breathing room. Have crusades from a rising Russia aimed largely at the Ottomans who are unable to finish Byzantium off immediately. They exchange rights to the city for perpetual control of Morea, (the approximate area of the Duchy of) Athens, Thessaly, Crete, Corfu, and most of the Aegean islands outside of Rhodes. Patriarch of Constantinople to relocate to Mt Athos as part of the deal and at the "request" of the Emperor, who reliquishes that title and is simply a hereditary office for "First of the Greeks". Ottoman forces have free reign of the nation militarily and are Ottoman in all but civilian leadership and name. Greek identity is kept alive over the long centuries and they bide their time until the late 1700s/early 1800s when they revolt and take most of modern Greece with them, allying with the Serbs and Bulgarians who win their freedom at the same time. They combine to wage a united series of Balkan Wars that allows Constantinople to come back into Greek hands along with much of northern and western Anatolia. The Serbs, already on very friendly terms with the Greeks, consent to a dynastic union while the Bulgars eventually merge their royal line into the mix as well. Romania becomes something of a satellite state and plays the Empire off of Russia, though the two get along fairly well (maybe it eventually joins too?). A thoroughly Orthodox nation emerges by 1880 running from Belgrade to Ardahan at the Georgian/Armenia border and the Danube delta to Dalaman (just across from Rhodes). Ottoman forces are humilitated largely in a Crimean-style conflict that sees the Empire tag team with Russia to being serious devastation to the Ottomans, the British and French are kept at bay only by the rise of the Empire as a newer counterweight. Its antagonism of Austria-Hungary is well-known by 1895 when it proceeds to come to the brink of war over Bosnia following several incidents along the Croatian frontier. This empire is not united by nationality but by faith, becoming a very Orthodox bastion that sees large areas of Bosnia and Albania convert by 1935. Ottoman forces still control Syria and the Levant with nominal control over Libya (taken by the Empire in 1902) and Egypt (British in all but name by 1860) but they are strongest in central/southern/eastern Anatolia which they retain. By the time of WWI there is palpable tension between Austria-Hungary and the Empire with the Ottomans aching for a chance at revenge and reconquest of the City of Men's Dreams...
