Byzantium was a continuation of the Eastern Roman empire that became more in touch with Greek culture over time, but how did the lower classes identify? I.e., would they identify as Roman, or Greek? Did it vary throughout time?
They called themselves Romans and their Empire the Roman Empire. When Constantine XI died in 1453, he died as Emperor of the Romans, not "Byzantine Emperor," as that is a later invention by historians both as a derogatory name and as a useful way to differentiate between the classical Empire and the Christian Empire of the Middle Ages.
There was some revival of a Greek identity in the twilight years of the Empire, mostly as a way to highlight the difference between the Greeks and the West. For example the work of Niketas Choinates describes how Theodore I of Nicaea fostered philosophy for the purpose of preserving it in the Greek sphere so that it wouldn't go the west. Other examples include how John Argyropoulos, one of the men who helped start the revival of classical learning in Italy, tried to pressure Consantine XI to not declare himself Emperor of the Romans, but as King of the Greeks. This identity, as I said earlier, was mostly to contrast themselves from the latins and their ever increasing influence on the Empire.
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