“Continuity between the Greek Roman Empire and the classical heritage needs to be emphasized because it bears on both Christian and Islamic civilizations. However, the word Byzantine hides this continuity. It is a word even less justifiable to designate the inhabitants of the Christian Greek Roman Empire of the Middle Ages than the word Indian is to designate the sixteenth-century inhabitants of the Americas or the word Iberia (now almost universally adopted among specialists in the English-speaking scholarly world) is to designate medieval Spain. The word Indian is an involuntary error resulting from an unavoidable lack of knowledge about an existing continent, but the words Byzantine and Iberia are artificial academic constructions resulting from ideology.”The German historian Hieronymus Wolf (13 August 1516 - 8 October 1580) was the first to use the term “Byzantine” — in his 1557 work Corpus Historiæ Byzantinæ — to label the later years of the Roman Empire. He got the term from "Byzantium", the name of the city of Constantinople before it became Constantine's capital. Montesquieu used the term “Byzantine” in his own works, but the term came into general use only in the mid-19th century. Here’s Fernández-Morera’s perspective on the origin of the term “Byzantine Empire”:
“In fact, the term Byzantine Empire was invented in 1557 by the German scholar Hieronymus Wolf, who as a Protestant would not have been sympathetic to Eastern (or Orthodox) Christians, to indicate that these culturally Greek people of the Eastern Roman Empire were not Romans, and somehow not even Greeks. His scholarly decision may also have been influenced by the fact that the Holy Roman Empire of Charlemagne and his successors had claimed the name Roman for itself… Eighteenth-century Enlightenment scholars such as Montesquieu, who despised Orthodox Christianity perhaps even more than Roman Catholicism, adopted the term, thereby emphasizing that these presumably retrograde Christian Greeks had nothing in common with those pagan Greeks admired by the Enlightenment. This artificial construction, Byzantine, already charged with Enlightenment-created connotations of convoluted formalism and corruption, has continued to be used by most Western historians.”I think that the term “Byzantine Empire” is a misnomer and a synecdoche; it does not tell us that this empire used to call itself “Roman Empire” during much of its history. However, Hieronymus Wolf was not at fault in using the term “Byzantine Empire” — after all, the capital of the empire was called Byzantium before the age of Constantine.