By refugee settlements I mean settlements like the medieval English settlement of New England after William the Conquerer. Another example would be like how Brazil settled Confederates after they lost the American Civil War. The only possible info about a settlement or colonial ideas was the theory that Christopher Columbus was a Byzantine prince.
I don't understand what you are asking... The "Byzantine Empire" died out with its last emperor Konstantinos Palaiologos the 11th when Constantinople fell in 1453 (actually it lived on for a year or two in Peloponnisos). The empire as a political entity ceased to be then. If you are asking if the Greeks engaged in colonial activities in the Americas, the answer is no. There were no greek colonies in the Americas.
The only possible info about a settlement or colonial ideas was the theory that Christopher Columbus was a Byzantine prince.
Columbus was a Genoese (like Italian) in service of Spain. Who told you you that he was Byzantine Prince? Find the guy and punch him.
EDIT: While I enjoy theories as much as the next guy, and being Greek myself, I cannot but to take pride on Columbus secret origins. But information is severely lacking and most fingers point to Genoa with no high noble ancestorship.
Also note to op: Never trust wikipedia.