Greek became the main language of the administration after the seventh century due to Heraclius (Emperor 610-641 CE). The switch was primarily because it was easier to run the Empire in Greek than in Latin. Even before the fall of the West, Greek was one of the most, if not the most, spoken language in the Empire, as it had been spoken for much longer in many provinces of Rome than Latin. After the lands acquired by Justinian I were lost, the lands that the Byzantines controlled had been speaking Greek a lot longer then they had been speaking Latin. This Wikipedia Article also explains this: http://en.wikipedia.org/wiki/Byzantine_Empire#Language

Wouldn't it have been more useful for them to speak Latin for prestige purposes? After all they insisted on being the (remnant of) the Roman Empire

The Byzantines were practical. After the Arab conquests lopped off Egypt and Syria, the empire was predominantly Greek-speaking. Even its southern Italian territories had been long settled by Greek colonists since before the rise of Rome.

Greek was also the language of Homer and the Septuagint and many of the Church Fathers, nothing to sneer at prestige-wise. In literature, writers would go so far as to affect a classical "Attic" style named for Attica, the home region of Athens, that would have been obscure to the people at large.