Taken from Wikipedia:
Yamauchi wanted primarily to target the North American market, so he
mandated that the game be given an English title, though many of their
games to this point had English titles anyway.
Miyamoto decided to name
the game for the ape, whom he felt to be the strongest character.[sup]
[19][/sup][sup]:39[/sup]
The story of how Miyamoto came up with the name "Donkey Kong" varies.
A
false urban myth says that the name was originally meant to be "Monkey
Kong", but was misspelled or misinterpreted due to a blurred fax or bad
telephone connection.[sup]
[26][/sup]
Another, more credible story claims Miyamoto looked in a
Japanese-English dictionary for something that would mean "stubborn
gorilla",[sup]
[20][/sup] or that "Donkey" was meant to convey "silly" or "stubborn"; "Kong" was common Japanese slang for "gorilla".[sup]
[15][/sup][sup]:238[/sup]
A rival claim is that he worked with Nintendo's export manager to come
up with the title, and that "Donkey" was meant to represent "stupid and
goofy".[sup]
[25][/sup][sup]:48–49[/sup] In the end, Miyamoto stated that he thought the name would convey the thought of a "stupid ape"