I've read that the Byzantines would refer to themselves as Romans up until the end of the Empire. I've also heard they didn't refer to themselves as the Byzantine Empire, and that term was given to them after they fell.
I'm on the origins of the word "Byzantine". And what the people of the empire would of called them selves through the eras.
The term "Byzantine Empire" and "Byzantines" are historical terms created hundreds of years after the end of the Empire.
The citizens of the Eastern Roman Empire saw their State as a continuation of the Old Roman Empire and referred to it as "Roman Empire" and called themselves "Romans".
From Kazhdan, Aleksandr Petrovich; Epstein, Ann Wharton (1985). Change in Byzantine Culture in the Eleventh and Twelfth Centuries. Berkeley and Los Angeles: University of California Press.