When referring to the Eastern Roman Empire in the modern day, we often call it the Byzantine Empire or Byzantium. While I know for a fact that the early Byzantine Empire still called themselves the Roman Empire, was there ever a point in history when they referred to themselves as the Byzantine Empire, or is that a name us modern day people came up with?
Also, for bonus points, where does the name "Byzantine" even come from?
"Byzantine" comes from Hieronymus Wolf's Corpus Historiae Byzantiae, a historical work composed in the 16th century, where he draws on the ancient name for Constantinople, Byzantion. Hieronymus Wolf was a German in the Holy Roman Empire, and as such wanted to portray his home as the true successor of the Roman Empire of Antiquity by rejecting the notion that the Eastern Roman Empire was truly Roman. Later French writers would reuse the term and it eventually stuck later on, as there was no Byzantine Empire to really contest.
The Greek-speaking peoples of the ERE self-identified as both Roman (Rhomaioi) and Greek (Graikoi) in that they acknowledged that their ancestors were the Ancient Greeks and that they were the living continuation of the Roman government. In fact, the debate over what the Greeks would call themselves was hotly debated around the time of Greek independence. The current Greek ethnonym "Hellene" was shunned by the medieval Byzantines, as it carried connotations of heathen beliefs (much as how the Latin word paganus, pagan, originally meant "rustic" but eventually came to mean "heathen", as pagan religions were considered the domain of rural yokels).
To sum it up, the medieval Byzantines never called themselves "Byzantine". The learned elite may know of their capital's past history and former name as Byzantion, and would at times call the capital that, but the empire itself was called "Basileia ton Romaion" or "Romania", or even "Politeia tes Romaion", which means Roman Republic.