The convention of referring to the medieval incarnation of the Eastern Roman Empire as Byzantium began with the German humanist Hieronymus Wolf. (You can read about Wolf's life and career in this chapter).

In 1557, Wolf published his Corpus Historiae Byzantinae, a pioneering edition and Latin translation of the "Byzantine" historians John Zonaras and Nicetas Choniates. To these he later added the chronicles of Nicephorus Gregoras and (someone else's translation of) Laonicus Chalcocondyles. A later edition, with all four authors, can be read here (if you happen to know Latin).

Wolf's decision to call these authors "Byzantine" as opposed to "Roman" stemmed from his contempt for the "Greek" Roman Empire and (at least equally importantly) for the "debased" Greek of the authors he translated. His sense of the period is clear from the lengthy subtitle of his Corpus, which (translated) reads in part: "the deeds of the Greek emperors from the time of Constantine the Great to the final Constantine and the occupation of Constantinople by the Turks..." Wolf, in short, was anxious to distinguish the "decadence" of Byzantium both from the political power of the Roman Empire and from the stylistic elegance of Classical Greek. "Byzantine" (the name, of course, of the Greek city that was refounded as Constantinople) was the term he seized upon to indicate that difference.

The seventeenth-century French series Corpus Scriptorum Historiae Byzantinae and Du Cange's synthesis Historia Byzantina (1680) established "Byzantine" as the accepted description for the medieval Eastern Empire. Edward Gibbon's extensive use of Du Cange in his Decline and Fall of the Roman Empire helped to establish the convention in English-language scholarship.

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D. Reinsch, "Hieronymus Wolf as Editor and Translator of Byzantine Texts." In P. Marciniak & D.C. Smythe, The Reception of Byzantium in European Culture since 1500 (Farnham, 2016), 43-53