The origin of the term 'Great' Britain.

As was said before the term "great" was meant as "larger" as opposed to "lesser" Britain (which is referred to as "Brittany" in English at the present time).
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It is actually a very old term, used perhaps hundreds of years before the Norman conquest. Many of the Britons crossed the channel to get away from the Anglo Saxon invasion and formed "lesser Britain". King James preferred to be called the King of Great Britain, rather than the King of England, Wales, and Scotland. Parliament at the time objected to the term, but ther king overruled him.
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By the time the parliaments were united, the term was accepted as the name of the nation. When Ireland was fully absorbed, "Great Britain" was replaced with the United Kingdom.
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Should Northern Ireland choose to adopt the Euro and unite with the Republic of Ireland in the future, the country may re-adopt the name, but is more likely to simply call themselves Britain.
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"Great" is derived from an old English word. "Big" is from roughly 700 years ago, and is probably of uncertain Scandinavian origin. "Large" is from a Latin word. In modern English the words "big" and "large" have assumed much of the meanings of "massive in size", and "great" generally takes on the meaning of "important". But the term "Great Britain" is from the Old English period in history. However, since it's formal usage corresponded to the rise of the British empire, most people equated "Great" with "Awesome" or "Important".