England were the dominant nation in the British Isles, what benefit did they have to rename their global empire the 'British' empire instead of the English empire? What is in it for them?
Simply put, when James I (James VI of Scotland) ascended to the throne of England, this was the formation of a personal union between the two kingdoms of England and Scotland. i.e. the two countries were ruled by the same monarch but internally were separately governed with separate legal systems, parliaments etc.
This was superseded with Acts of Union in 1707 when the Scottish systems were merged into the Parliament of Westminster. At this point, there ceased to be two distinct legal and political entities of England and Scotland and henceforth were the Kingdom of Great Britain. So named after the geographical entity Great Britain, the larger of the two islands that make up the British isles (the other being Ireland).
It was called the 'British Empire' because it was operated by and for the benefit of the peoples of the (United) Kingdom of Great Britain and to a lesser and more nuanced extent the peoples of Ireland. It was not only the English who colonised, ran the government, were peers of realm or elected to the Commons. Scots, English, Welsh and Protestant Irish all partook in the enterprise.
A History of Britain : The British Wars 1603-1776 / Simon Schama. London : The Bodley Head, 2009.
Hey OP, could you possibly clarify: Are you asking when the name "Britain" became applied to the British Isles (with Great Britain for the area that's modern-day England, Wales and Scotland) or are you asking why the political entity of Great Britain was named Great Britain?