Well Suraknar, Im not going to discuss this thoroughly, but here's my opinion anyway.

If you would ask a man in Medieval Europe about the Byzantine Empire, he'd look at you and tell you he doesn't know such place. He probably doesn't know about a city called Byzantium either. When you start about Greeks he may get it already and when you call it the Roman Empire he'll know what you're talking about.

The name Byzantines, although justified, is made up 500 years after the empire ceased to exist. It's a little ridiculous to call your own empire - You do play the emperor - as if you lived some centuries later... Even if they knew Byzantines was a more appropiate name, they would never use it, as they had a good reason for keeping the Roman name:
The Emperor still had important claims on west europe. He was protector of Rome, the king of the Franks was his vassal, the Norse would come to Constantinople only because the legendary Romans lived there.

The westerners, however, called the Byzantines the Greek Empire, as they already had their own Roman one: the Holy Roman Empire. The HRE did claim absolute power over all the previously roman country, like the Byzantines did. If the Byzantines would give up their name and with that their claims on the west and their legendary stories, it would do severe damage.

... And that's why I think it's a little silly to call them Byzantines. If you really want to make some distinction with the Roman Empire you could call them East Romans, or the Greek Empire (but that's a western name, so if you do that you should call all the other places as the English called them).