Apache, the HTTP server and software foundation, was apparently chosen as a name for several reasons.
The Apache website's FAQ section said it was "a cute name that stuck" adding: "Apache is "A PAtCHy server". It was based on some existing code and a series of "patch files"."
Brian Behlendorf, one of the creators, would later admit they loved the fact it was a pun on a patchy web server and Apache, the mighty North American Indian people. "It just sort of connotated: "Take no prisoners. Be kind of aggressive and kick some ass" he said.
However, the Apache Foundation would later state on its website that "the name 'Apache' was chosen from respect for the various Native American nations collectively referred to as Apache, well-known for their superior skills in warfare strategy and their inexhaustible endurance. It also makes a cute pun on "a patchy web server" -- a server made from a series of patches -- but this was not its origin."
Sources: Apache FAQ | Apache Power interview