Nobody really knows for sure.  It's been suggested, and sounds plausible
(but there's no real evidence), that it was called F-19 to start with, but
the number was changed as a security measure after the open press started
using that designation in the early 1980s (the aircraft first flew in 1981,
but wasn't revealed to the public until 1988).  Why they picked F-117 as
the new number is a mystery; there are three main theories, any of them
fairly plausible.

The first theory has it that the "stealth fighter" (actually it's a bomber;
see below) was flying from the same bases as the small fleet of captured
Russian aircraft that the USAF flies; these are believed to use the
nonexistent designations "F-112", "F-113" and so on as a cover, and the
F-117 just happened to be the next number in sequence.

The second theory claims that the aircraft was using the call sign "117"
(possibly for reasons connected with the above, or possibly just an
arbitrarily assigned number) on some of its early test flights, and the
number just happened to stick (presumably for lack of any other
designation); when Lockheed got around to printing pilot's manuals for the
aircraft, they were labelled "F-117", and from then on it became official.

The third theory is that there isn't any reason; the Pentagon just picked a
number at random.

The mythical "F-19" may have been part of a "leak identification" project;
it's common practice in many "black" projects to create several false
stories and track down leaks by watching to see which one gets out.

There's also the separate question of why it was given an F-series
(fighter) designation at all, when it's clearly a light bomber with
essentially zero air-to-air capability; it should have an A-series (attack)
or B-series (bomber) number.  Again, the Pentagon isn't telling, but a
favourite theory here on the Net is that the USAF, being dominated by
former fighter pilots, couldn't bear the idea of its most glamorous plane
having anything but a fighter designation...

The F-117 has been popularly known as "Nighthawk" for some time; the Air
Force made the name official on 24 June 1994.