The best POD for the Byzantine Empire to survive until at least the 17th century would almost certainly be to have the Byzantines win the Battle of Manzikert.
Then if the Byzantines can follow up on this win and successfully defend Anatolia from the Turks in the 1070's while stabilizing their society one way or another, they'd likely be able to retain most if not all of their holdings that they had in 1071 for a while longer.
I could even see a stronger Byzantium at some point seeking to retake the Holy Land from the Muslims for some reason.
Of course, having a stronger Byzantium post-1071 AD might lead to further clashes between the Vatican and the Byzantines as the former would be unhappy that such a powerful empire dared to defy them in terms of religion and seemingly get away with it. It could even lead to wars between Byzantium and other European Nations who keep sending Crusading armies to try and "free" the Holy Land from the Heretical Byzantines.
It could of course be possible that the Vatican and Byzantium work out their religious differences as at the time, no one thought that the schism between Western and Eastern Christianity would be centuries long like it was in OTL but I'm just stating something that might happen if that isn't the case.
When the Mongols DO arrive in the 1200's, I could see them engaging in some clashes with the Byzantines at first only to be paid off by them at times (the Byzantines did this with barbarian tribes back in the 4th century and beyond). Still, the Mongols might try to take Constantinople as it would almost certainly still be a super-wealthy city by this point but I can see the Byzantines successfully fending them off if that were to happen.
If they had survived as a strong empire till the 15th century, then I could see the Byzantines sending expeditions to the Indian Ocean for trade and perhaps even set up colonies only to end up clashing with the Muslims and Portugal.
Whether or not European nations discover the New World on time depends on their relations with Byzantium. If they have good relations with them and are allowed to use their lands to engage in trade, then the discovery and development of the New World is delayed as they wouldn't have as much incentive to find new routes or develop new lands. IF they have as poor relations with Byzantium as they did with the Turks on OTL, then nations such as Spain and Portugal will likely send expeditions to find other routes to the Far East only to "discover" the New World.
Either scenario would make things QUITE interesting but in either one, Byzantium would definitely survive as a major powers until the 17th century.