The term 'Byzantine Empire' is a Western invention in its entirety. Straight up until 29 May 1453 they called themselves Romans and their state the Βασιλεία Ῥωμαίων (Basileia Rhomaion/Roman Empire).

What then is the origin of the terms Byzantine and Byzantine Empire? The original name of the city of Constantinople was Byzantion. You might occasionally find references to the city itself using this nomenclature (in the same way that today you can find romanticized references to Paris as Lutece or as Paname), but never the Empire as a whole. The first application of the term 'Byzantine' to describe the totality of the state was by the German historian Hieronymus Wolf in his Corpus Historiae Byzantinae, published in 1557. In the 17th century, several French historians such as Du Cange also applied the term 'Byzantine' to the empire, but really it was the later work of more popular figures writing on the subject such as Montesquieu (in France) and Gibbon (in Britain) which led to 'Byzantine' becoming the predominant term referring to the Eastern Roman state in the 18th and 19th centuries. Getting away from the realm of pure history, I would make the argument that this was done mainly because of a desire to separate what was seen as a decadent, despotic Greek empire from the perceived glories of classical antiquity on the part of early modern European scholars. Gibbon in particular is incredibly dismissive of Byzantium, more or less describing its thousand year history as one of decline and decay.

It's worth pointing out that this naming is primarily a Western process, however. Ottoman sources referred to the ERE as 'Rûm' and its own Orthodox Christian population as 'millet-i Rûm' (the Roman nation) more or less through the entirety of its existence. Indeed, when Mehmed II took the city of Constantinople he accorded himself the title Kayser-i Rûm (Caesar of Rome). Greeks paralleled this--you can find the modern Greek language being referred to as Romaiika (derivative of the term Romaioi which Byzantine Greeks used to refer to themselves) as late as the 19th century.

Beyond the obvious popular comprehensive sources on the ERE in English such as John Julius Norwich, you could check out Nadia El-Cheikh's Byzantium as Viewed by the Arabs for the Turkic perspective on the empire. Rosser's Historical Dictionary of Byzantium also offers a fairly broad overview of the etymology behind the empire's various nomenclatures.