Turns out I'm wrong about the origin of the term "Byzantine" or "Byzantium" - the term was apparently coined by Montesquieu in the 18th century, since he didn't want to refer to the empire by the term Roman or Greek, since those were "classical" cultures which were immensely nobler than the decadent, dishonest "Byzantines." Notably, his ideas were the basis for some of our founding fathers, who saw themselves as heirs to a classical tradition, which thinkers like Montesquieu had distinguished from the "Byzantine" empire.
Professor Clifton Fox argues persuasively that the empire and its citizens referred to themselves as Romaioi, and thus the empire really ought to be referred to as the "Romaion Empire" from the Greek "Basileia Romaion" [Empire of the Romaioi]. Essentially the empire was a recasting in Greek of an Italian empire that was modeled on a Greek original. Clear as mud? Calling it the "Byzantine" empire because Constantinople was originally called Byzantium makes about as much sense as scholars 1,500 years from now deciding to call the scholarship on the American Congress Jenkins Hill studies because that was the name of Capitol Hill before it became the seat of the U.S. legislative body. (The analogy would fit better if there had originally been another name for Washington, but I don't believe that there was). The term Romaioi is a somewhat corrupted version (translated to Greek) of the original Latin Romani, in the same sort of way that London is derived from Londinium in the original Latin.
Anyway, at least now I have a slightly more historically accurate title for the culture - it's bothered me that we're referring to a thousand years of history by a term that the inhabitants wouldn't have recognized. Seems like they deserved better than that. They're Romaioi.