I could have sworn that within contemporary discourse it was always called the Roman Empire, but at a certain point it became easier to refer to those lands in the East as Byzantium rather than Roman. I was curious when that change occurred.
During the existance of the Empire, the Byzantines themselves referred to themselves as Roman and the Roman Empire (usually "Rhomanoi" since most were Greek-speaking).
The western christian usually referred to the Byzantine Empire as "the Empire of the Greeks" or "the domain of the Greek Emperor" or "the Empire of Romania" (taking the Greek name for Romans) after the Pope had crowned Charlemagne Holy Roman Emperor. The catholics considered the Holy Roman Empire the real successor of the Roman Empire, since it was catholic, and did not refer to the Byzantine Empire as Roman.
The muslims usually referred to the Byzantine Empire is "Rhum" or "the Empire of Rhum", a transliteration of Rome. This is also why they called their central Anatolian realm "the Sultanate of Rhum" (or Rûm).
The term "Byzantine Empire" (after the Greek city Byzantion, which was renamed Constantinople, expanded and made capitol of the Roman Empire by Constantine the Great) was first recorded in 1557, when the German historian Hieronymus Wolf used it in a historical work. However, it was not before the 19th century that the term came into widespread use.
It started circa a century after the end of the Empire.
in the Early Middle Ages the Eastern Roman Empire was THE Roman Empire, the one and only.
with the rise of the Holy Roman Empire in the West, and especially with the Great Schism of 1054, in Western Europe it was no longer considered the legitimate continuation of the Roman Empire but an "Empire of the Greeks". The term Romania sometimes appears to designate the Byzantine Empire.
After the fall of Constantinople in 1453, historians tried to separate the Ancient Roman Empire (whose glorious past modern Europe wanted to emulate) from its Mediaeval continuation. German Hieronymus Wolf was the first to refer to it as the Byzantine Empire in the Corpus Historiae Byzantinae in 1557.