The usage was coined by a German historian named Hieronymus Wolf, who lived in the 16th century. From what I know about him, he was interested in Greek history and literature so perhaps he chose "Byzantine" for that reason, but honestly I'm not sure of the reason why he specifically picked "Byzantine."

Westerners contemporaries of the Byzantine Empire gave it various names to distinguish it from "their" Holy Roman Empire, which they saw as a truer successor. "Romania" was one term as was "Empire of the Greeks." During Byzantium's stronger centuries, accrediting ambassadors to the "Emperor of the Greeks" was a colossal faux pas and could lead to those ambassadors being thrown into prison or expelled.

As far as I know, though, the term "Byzantine Empire" completely postdates the empire.

Even though the city was renamed, the Byzantines themselves would call it "Byzantion" from time to time. Stylistically it was popular to make classical references in literature, so the Byzantines would refer to their city as Byzantium, the Turks as "Persians" since they came from more or less the same area, anyone coming from the steppes were "Scythians" and so on. The Byzantines didn't extend "Byzantion" to cover their whole state, though, like modern historians do.

Follow-up, would the demonym be Constantinopolitan? Was that term ever used historically?