What does Byzantine mean?

"Byzantium" is a Latinization of the Greek Word.

The name of Βυζάντιον (Byzantion) is believed to be of Thracian or Illyrian origin and may be derived from a Thracian or Illyrian personal name, Byzas.

Ancient Greek legend refers to a legendary king of that name as the leader of the Megarean colonists and eponymous founder of the city.

Byzantium is one of the older names for the city now called Istanbul in Turkey.