There were Celtic people called Brittos. The Romans called the land Britannia. Once England, Wales and Scotland united, the term "Great Britain" came into use. The United Kingdom of Great Britain and Northern Ireland is the full modern term. (It used to be just "and Ireland" till they chucked us out in the 1920s).
The Act of Union (1707) created the United Kingdom of Great Britain. With the Act of Union (1801) in became the United Kingdom of Great Britain and Ireland. which in turn became the United Kingdom of Great Britain and Northern Ireland in 1921 after the establishment of the Irish Free State. James I & Vi was the first to use the term Great Britain but this did not refer to a political state. He also referred to Scotland as North Britain and England as South Britain.
Certainly an early Celtic tribespeople were called Britons and a Latin version of this was Brito or Britto.
These islands were described by Tacitus in around 55BC as 'The land at the extremities of the earth'.
Four centuries earlier Heredotus had called them 'The Tin Islands' a little more complimentary.
The name extended across the channel when the Celts settled in Brittany.
Quite why they were called Britons I don't know, but many claim it may have come from a chieftan called Brutus.
Don't forget Little Britain, known as Brittany. The people are Celts, who settled there from Britain in around the 5th century AD to join the Celts already there.