Others place it yet further in 476, when the last western emperor, Romulus Augustus, was forced to abdicate, thus leaving sole imperial authority to the emperor in the Greek East.
In any case, the changeover was gradual and by 330, when Constantine inaugurated his new capital, the process of further Hellenization and increasing Christianization was already underway.
The name Byzantine Empire is derived from the original Greek name for Constantinople, Byzantium. The name is a modern term and would have been alien to its contemporaries. The Empire's native Greek name was Ῥωμανία Romanía or Βασιλεία Ῥωμαίων Basileía Romaíon, a direct translation of the Latin name of the Roman Empire, Imperium Romanorum. The term 'Byzantium' seems to have been first re-introduced by 15th century classicising Greeks who preferred it to 'Constantinople'. Through the translations of their texts into Latin, its usage was picked up north of the Alps by historians who were just becoming acquainted with the art of historiography. Hence, to the best of our knowledge, the term Byzantine Empire was introduced in 1557, about a century after the fall of Constantinople by German historian Hieronymus Wolf, who presented a system of Byzantine historiography in his work Corpus Historiae Byzantinae in order to distinguish ancient Roman from medieval Greek history without drawing attention to their ancient predecessors. So far, it appears that there has been no study tracking the reasons why that term came to gain prominence.