From what I understand, the label "Byzantine Empire" is a term concocted in the 16th century. Why do we still use that term when the "Byzantine" Empire is a direct continuation of the Roman Empire and its contemporaries (foes and citizens alike) still called it so with the notable exception of the European states who try to claim the "Roman Empire" mantle for political legitimacy?
It seems like a rather misleading distinction to me. Is it common to re-name empires when they lose a significant amount of territory?
Honestly, it is kind of unavoidable. A distinction between the Roman Empire of Antiquity and the Roman Empire of the Middle Ages (the pivot point being 700 or so) is useful and necessary. Once you accept that, you will need new terminology, and because the Roman Empire was around roughly equal periods before and after 700 the term "Late Roman Empire" doesn't make any sense and, that aside, is already taken by the period from about 300-500 (or 6/700). So you will basically be forced to say either "Medieval Roman Empire" or "Byzantine Empire", both of which are Eurocentric impositions.
Historians still usually use the term "Byzantine Empire", but are highly cognizant that it is a deficient and deceptive term.
A basic justification of the term "Byzantine" is to distinguish a Rome-centered empire from a Constantinople-centered empire.
Emperor Constantine founded the city of Constantinople on the site of an earlier, pretty ancient Greek city called Byzantium or Byzantion, and Constantinople soon became big, rich, populous and the capital of the Eastern half of the empire. After Rome fell to the barbarians for good, Constantinople was the seat of the empire for the rest of its history. This shift reflects a number of structural differences between the new and old empire, such as the shift from Italian to Greek ethnic and linguistic dominance. Constantinople also had less pre-Christian, pre-monarchist baggage than Rome had, and the shift reflects a severance of the pagan, republic past (at least to many contemporaries).
Thus, 'Byzantine' distinguished this radical relocation of capital. Of course, 'Constantinopolitan' would be more accurate for this purpose, but is also a ridiculous mouthful. Also, the term "Byzantium" or "Byzantine" was used by contemporaries though ONLY in reference to the actual city of Constantinople itself or its full-time inhabitants, not to the whole empire as a whole. As has been said, they generally called themselves Greeks or Romans.
Basically, Medieval Roman Empire would probably be the best as Tiako said, and the term Byzantine has its origin mostly in the Western Europeans historically wanting to deny the empire the legitimacy of being a Roman continuation and claim such legitimacy for the catholic church, the German empire, etc.