This is from traceit.com. I can't speak for its veracity but you can continue the search yourself given this info, i bet.
"The Everest family can trace their ancestors back to the ancient territories of England between the 11th and 12th centuries.
The Everest family traces their ancestral roots back to Norman origin, and first appeared in ancient medieval records in Herefordshire .
That from very early on the Everest family not only held lands and estates in England but were also actively allied with other influential families. They also branched out into other territories and holdings, before taking the long voyage to the new world. "