This honestly doesn't change that much as far as I see it.
Firstly, while Constantine XI died without any children, his brothers Thomas and Demetrios Palaiologos continued to rule the Morea until 1460. Thomas and Demetrios both had children (although Demetrios only had a daughter), and while Demetrios stayed in the Ottoman Empire, Thomas moved to Italy and lived on a pension from the Pope until he died in 1461.
Thomas' descendant, as mentioned above would continue to intermingle in Western Europe until the 17th Century when the direct line runs out.
Alright so let's assume that the direct line doesn't die out, and that direct Palaiologi continue to exist in Western Europe.
By the time the Greek War of Independence happens (I assume this POD doesn't really cause that many butterflies, it's pretty minor), the remaining Palaiologi will have already been assimilated into the countries they have moved to, and the Great Powers will want to pick a monarch they can have influence over in Greece, not some family of nobodies.
The prospect of any Palaiologi surviving in the Ottoman Empire itself I find pretty unlikely though. Only look at what happened to the Komnenoi IOTL. David Komnenos was arrested and executed on trumped up charges, his sons shared the same fate and his daughter was put in the Imperial Harem.
Having a legitimate symbol of Roman resistance wandering around, owning estates in the Ottoman Empire was highly inconvenient and risky. Especially a hypothetical direct descendant of the Marble Emperor himself.