Guys, i'm afraid the OP asked for PoD's and scenario proposals that sound outlandish but had a plausible chance to happen, not for OTL events that would sound ASB to an ATL inhabitant's ears. The latter has already been covered in multiple threads here.
Fair enough. Let's see:
1) More successful
Latin Empire. The Fourth Crusade by itself was bizarre enough IOTL, and its byproduct surviving might be even more bizarre, but not impossible. Perhaps if the Venetians did not get Baldwin of Flanders chosen as Emperor, but rather the more well-connected and popular Boniface of Montferrat, we could have the "Empire" born with even more territory, comprising Thrace and Thessalonica. In any case, they would need to defeat the Nicean remnant ASAP and water down the constant Bulgarian invasions. Afterwards, reinforcing the suzerainty over the various fiefs of the
Frankokratia would be much easier and permit a chance of medium-term survival.
2) A real existing
Nova Anglia (something I really wanted to see in a TL). Anglo-Saxon refugees fleeing from the Norman Conquest voyage to Constantinople, join the Varangian Guard (this really happened), and the grateful Komnenoi emperors grant them a fief in Chersonesus (how we get there from point A to B, I don't know), becoming a sort of early Theodoro, a small city-state focused on trade, likely a republic, adopting Byzantine culture and language, which survives by simply being too insignificant for the various nomadic invaders in the Steppe to care about.
3) France actually remaining a (functioning) Republic, likely federalized, after the Revolution, without all the Napoleonic shenanigans. I've seem some people argue that France would inevitably fall back into monarchism, sooner or later, even if not under the Bourbons, or be captured by a succession of military dictatorships under the guise of republicanism, but I'd like to think that it would be indeed possible to see the First Republic survive in certain circumstances.