I heard that the word 'Byzantine' is a modern way of referencing the Eastern Roman Empire after the West fell. What did the Sassanids refer to them as? What did the Arabs and Europeans call them? When Constantinople was conquered, did the Turks yell in joy and proclaim they finally destroyed the Roman Empire or did they call it something else?
The latin sources generally speaks of the greeks and of the land of the greeks but I honestly don't know if they used another term. The arabs and the turks called the byzantines rumi (romans), a word that derived to means the infidels in general. Their country was simply Rum, that's why there was a Sultanate of Rum in Anatolia after the total turkish victory of Manzikert.
My favourite part is the last one. After the conquest of Constantinople Mehmet II used the title of Kaisar-i-Rum, so he consciously reused a byzantine title, even if it's not really the main title used by the byzantine emperors (basileus autokrator). With that title he proclaim his autority over all the territory of the byzantine empire, which was true considering his extension in the late middle age
Ibn Battutah in his Travels refers to the Anatolian peninsula as "al-Rum" (Rome) and the territory controlled by the Byzantine Emperor specifically as Great Constantinople. I don't have access to the original Arabic, but I assume that would be al-Qunstantiniyya al-Kabir or something like that. So, either Constantinople or Rome, depending on whether the context is political or geographic.