The world's highest mountain is named for a man who said he did not want the honor.
Andrew Waugh, Surveyor General of India, reported the peak's height in 1856. Before that it was known by survey labels, including "Peak XV." Wikipedia says Radhanath Sikdar was the first to identify it as the world's highest peak, in 1852, and in March 1856 Waugh announced a height of 29,002 feet (8,840 m). Waugh recommended naming it for his predecessor, Sir George Everest. Wikipedia reports that Waugh reasoned that many local names existed, so favoring one would be difficult.
Everest did not welcome it. In 1857 he told the Royal Geographical Society that "Everest" could neither be written in Hindi nor pronounced by "the native of India." Wikipedia's article on him adds that he felt he had nothing to do with the mountain's discovery, and that he never saw it. He had resigned as Surveyor General in November 1842.
The Society adopted "Mount Everest" officially in 1865 anyway. Wikipedia notes that the modern pronunciation differs from his own. Sir George said his name as "EEV-rist," while most people now say "EV-rest."
The mountain had names long before. Wikipedia gives the Nepali name Sagarmatha, roughly "head in the sky," and the Tibetan name Qomolangma, meaning "holy mother." The Library of Congress Maps blog says the earliest known map to name it is the 1721 Atlas of the Whole Imperial Territory, which labeled it "Qomolangma." So the European name arrived more than a century after a written local one. Wikipedia adds that the name reached the West in 1733, on a map by the French geographer D'Anville that was based on that atlas.
As for the surname, Etymonline says name-books treat Everest as a variant of Devereux, a Norman name from Evreux in France.