Names are funny things, the name in question has been used for the entire archipelago, in one form or another, for thousands of years. Maybe part of the problem is the casual use of Britain to mean "The United Kingdom of Great Britain and Northern Ireland", it doesn't help that most people who live in Great Britain don't realize that the "Great" just refers to it being the largest island in the archipelago.

Personally I think that the UK should be separated back into England, Wales, Scotland, Northern Ireland, and Berwick-upon-Tweed. They can then individually decide whether to beg to be allowed back into the EU. The Windsor née Saxe-Coburg-Gotha clan should become private citizens like anyone else (if any country will have them)

On a more serious note, I feel that a lot of problems would be resolved by dividing much of Europe into smaller polities