I think part of the reason people find the Byzantines so interesting is because they have been marginalized in traditional historical narratives. Their power vanished pretty much completely, so there was no one to speak up for them for centuries. Not the Near East, which used them only to burnish their own "new improved" version, and certainly not the West (see: Gibbon). The Ottoman's have successor states to this day. Not so much the Byzantines.
What is really fascinating is the gap between the attention they get (really, how many mentions do they even get in school books?) and the tremendously important role you find that they played. The history of Byzantium (and Islam too) is the true history of the medieval period but here in the West we get both official history and popular myth all about knights and feudalism and zero mention of them. (Ever see a movie set in the Byzantine Empire? Neither has anyone else.)
You begin to see that the story you've been told is just a spin-off of a much larger one just off to the side out of you view. The Main Event. Byzantium.
Then there is the conditional about it. Manzikert may have become a AH cliche, but it is true. Also Yarmouk and Myriokephalon. As you research you begin to see that the propaganda about 11 centuries of continual, inevitable decline is crap. That the there are so many cases where the Empire was on the edge and could have come out better (or worse) and that nothing was inevitable about its demise. (Heck, if Morea and/or Trebizond had better leadership, either might still be around today even given 1453.)
Then there is the longevity of the thing and its Indiana Jones-like existence. The Roman state lasted 2000 years! The empire, 15 centuries! Generally speaking AH fans are big on the whole idea of longevity. What better than a neglected Empire that really was the actual, unambiguous continuation of prestigious Rome that is only seen as something different because of an accident of modern misnaming?
Even the Ottomans felt this way. Mehmet the Conqueror and his successors always maintained that they were the legitimate heirs of Augustus in a direct line. The Emperors of Rome. (As they saw it, trading Christianity for Islam was no different than trading Paganism for Christianity. And Turkish for Greek at court? No different than changing from Latin to Greek before.)
(And, OK, yes, the fact that the name has a "z," a "y," and a "ium" in it probably helps for a lot of people.)