Washington is a town near Newcastle, in the region of Tyne & Wear in the north-east of England.
In 1183 in the village of Wessyngton, England, a family took the name Washington. A descendent of that family moved to another country a few hundred years later and became quite famous, apparently.*
The name Washington evolved from its Anglo Saxon roots of Wessyngton, through a number of different spellings before settling as Washington.
Footnote: The first President of a United America was named George Washington after his ancestors who had lived in Wessyngton, and in turn the Capital of the modern United States of America was named after their first President. But hardly anyone remembers poor little Wessyngton in Tyne & Wear.