In the fifth century BCE a man called Byzas led a colonisation party from Athens to a peninsular on the eastern end of Thrace. The city thus founded was called Βυζαντίον/ Vyzantion as it is pronounced in Koinê and modern Greek, or Byzantium in the Latinised form.
In 330 CE Emperor Constantine (I / the Great) adopted the city as the new capital to replace Rome, and changed its official name to Konstantinopolis. Yet this name never wholly supplanted Vyzantion, and so the Roman Empire became the Byzantine Empire, right? Well, no. The Roman Empire remained the Roman Empire to its residents to the end in 1453. So where does this adjective and title come from?
The ideological roots of it go back to the turn of the ninth century when the Papacy, seeking to increase its power, usurped the Roman imperial title and conferred it upon Charles I of France (Charlemagne). The growing gulf between eastern and western Christendom, exacerbated by the schism in the churches and the atrocities of the Crusades, merged into the cultural ideology of Renaissance as Italians and others sought to recover their Roman heritage from the degenerate and perfidious Greeks.
As Classical Studies became established as an academic discipline from the Eighteenth Century, this adversarial ideology pervaded Western scholarship under the mask of scholarly detachment. Thus, western scholars needed a term to distinguish the real Roman Empire, from the unimportant portion of that just happened to survive for a thousand years beyond the fall of the Western Empire.
For me, it is the very survival of the eastern empire that makes it so important. The Roman Republic and early Empire prospered only so long as there were people it could conquer and loot. Once firm geographical limits were reached, decay set in. In contrast, the eastern Empire survived for as long again in the face of repeated and brutal assaults from East, North and West, showing a degree of resilience unparalleled by any other society in recorded history.
For these reasons I have rejected the perjorative term Byzantine, and embraced the terms these remarkable people used for themselves, mindful of their Roman heritage:
Ῥωμάνια / Rômania: this term was in use from the First Century CE for the territory governed by the Roman Empire
Ῥωμαίκοϛ / rômaikos: Roman, a Roman. Anglicised to Romaic
and, of course, simply Roman and Eastern Roman
In one area traditional academic practice remains. The era following the transfer of the capital is divided into three periods early, middle and late Byzantine. This practice is harder to revise, and so these terms will occasionally still occur.