It means to fly as Near As Possible to the earth .. NAP is an acronym. ... [You] should have asked an Army pilot first.

You almost caught us napping, BlackhawkAviator.

Good one! We extend a special welcome to any "Stranger" who can test our wits as ingeniously as you have.

Of the several kinds of "nap" in English, the most well-known is "nap" meaning "a short sleep" (or, as a verb, "to take a short sleep"). Modern English inherited this "nap" from the Old English "hnappian," meaning "to doze," but its ultimate origin is unknown. But that's OK, because the "nap" in "nap of the earth" has nothing to do with dozing. You're on the right trail when you assume that it is connected to the "nap" or surface of a carpet or cloth. This kind of "nap" arrived in English around 1440 from the Middle Dutch "noppe," meaning "tuft of wool." The "nap" of wool or cloth is the layer of projecting fibers on the surface, and "nap of the earth" metaphorically likens the hills, valleys, trees and so forth of the earth to the "nap" of a carpet.

While I suppose that any small aircraft would be capable of flying "nap of the earth," it seems to be primarily a helicopter tactic, and I have been unable to find the term used earlier than the Vietnam War.

The Word Detective, December 2002

http://www.word-detective.com/122002.html

If "nap" had, indeed, been an acronym for "near as possible", the expression would be "nap the earth", not "nap of the earth".