Certainly the Byzantines themselves never thought of themselves as anything other than the Roman Empire--there is a (potentially apocryphal) story of Charlemagne sending correspondence to Empress Irene and referring to himself as "Emperor of the Romans," to her great confusion and irritation: she was the Empress of the Romans, after all. Their understanding of themselves as being Roman is reflected in their interactions with the growing Muslim world--the Arabic name for what we call "Byzantine" is "al-Rum," "the Romans," while the Franks and other western Europeans who came to the Levant during the Crusades were called "al-Ifranj," or "the Franks." Furthermore, the Sejulq and other Turkish tribes moving into Anatolia established political entities including the Sultanate of Rum, because, as far as the Turks were concerned, the Byzantines were the genuine Roman article, and always had been. Historians in the West began referring to the later entity as the Byzantine Empire as the field of Late Antiquity began to develop further, especially in the 19th century, and primarily as a means of clearly showing a difference between the more ancient Rome which ruled the entire Mediterranean and the later entity which considered itself Roman and called itself Roman but was Christian, didn't control the original city of Rome, and had a culture that was primarily Greek rather than Latin.

Incidentally, the actual division between "Roman" and "Byzantine," the line that divides the two, is still a major argument among historians. When my Master's class and I were taking our comprehensive exams, one of my friends was actually asked "At what point can we truly mark a division between the Roman and the Byzantine Empires?," since there are so many potential places for that division: Constantine building Constantinople, Theodosius II dividing the Empire into Eastern and Western for the last time, the defeat at Adrianople, or the "fall" of the Western Empire, just as a few potential options.