It has everything to do with the Great (or General) Trigonometric Survey and its methodology. The Survey was a British initiative aimed at charting the physical geography of British India, a major part of which was measuring the exact height of mountain peaks as well as naming them. With the Himalayas and Karakoram mountain ranges, they first measured height and referred to peaks by abstract figures; in many cases, they decided to figure out local names later. Because there were several teams working on the same peaks at times, and from different angles, this nomenclature occasionally changed around. In the case of Everest, it was first designated 'peak b' by John Armstrong, after discovering that it was taller than Kangchenjunga, previously considered the tallest. Michael Hennessy, one of the other surveyors, had been using roman numerals to designate the peaks in his range, and called it 'peak XV'. It was the latter that Andrew Waugh, the Surveyor General in charge of the Survey, used in his report to the Royal Geographical Society in 1856. Wikimedia actually has the opening pages of that report, which are worth a read.
Now, the naming procedure in principle meant that they'd use local names to use in British maps, which was fine in the case of most peaks visible from populated areas. In the case of Everest, Waugh claims that, because it either didn't have a local name, or that such a name "will not very likely be ascertained before we are allowed to penetrate into Nepal and to approach close to this stupendous snowy mass" (Waugh 1857: 346). Hence, the peak was termed 'Everest' in honour of Waugh's predecessor.
As a related aside, the Karakoram range was so inhospitable and sparsely populated in some areas that one of its peaks actually didn't have a local name - because the peaks in the range were designated 'K' (for Karakoram) and a number, K2 still is the proper name for it.
Sources:
Gillman, P. (ed), Everest – The Best Writing and Pictures from Seventy Years of Human Endeavour. Little, Brown and Company, 1993 - The introduction has a great short overview of surveying history.
Waugh, A.S. 'Papers relating to the Himalaya and Mount Everest' in Proceedings of the Royal Geographical Society of London, Vol. 1 (1857), pp. 345-351 - Waugh's original paper naming Peak XV 'Mount Everest'.
Waugh, A.S. 'On Mounts Everest and Deodanga' in Proceedings of the Royal Geographical Society of London Vol. 2, No. 2 (1857 - 1858), pp. 102-115 - Discussion on a rival designation for Everest as 'Deodanga', where he shows that another survey team mistook Peak XIII for XV.
