Why was the Lockheed F-117 Nighthawk designated as a Fighter aircraft, and not something more appropriate, like an Attack or Bomber aircraft?

Historically, the USAF has referred to combat aircraft as either fighters or bombers. For example, when the USAF became independent in 1947, existing USAAF Pursuit aircraft were redesignated as Fighters (P-51 Mustang became F-51 Mustang). Existing USAAF Attack aircraft were either redesignated as Bombers (twin-engine, medium aircraft like the A-26 Invader became the B-26 Invader) or Fighters (single-engine, smaller aircraft like the A-24 Banshee became the F-24 Banshee). Of course, there are a few exceptions, like the A-7 Corsair II and A-10 Thunderbolt II.

An official USAF document, AFI 16-401 has the following descriptions:

A (Attack)—Aircraft designed to find, attack, and destroy enemy land or sea targets using conventional or special weapons. This symbol also applies to aircraft used for interdiction and close air support missions.

B (Bomber)—Aircraft designed for bombing enemy targets.

F (Fighter)—Aircraft designed to intercept and destroy other aircraft or missiles. Includes multipurpose aircraft also designed for ground support missions such as interdiction and close air support.

So, there is some overlap with the Attack and Fighter missions.

For the F-117 specifically, there are still a lot of misconceptions. Designation-systems.net has the best explanation that I have seen:

Although it is commonly called the Stealth Fighter, The Nighthawk should have received an "A" designation. It has no air-to-air role whatsoever, and the "A-for-Attack" designator is in fact prefectly made for tactical ground-attack aircraft like the F-117. The topic has been discussed to death among enthusiasts, but there seem to be essentially two possible reasons for the assignment of an F-designation:

• For the U.S. Air Force, all dedicated combat aircraft are either "fighters" or "bombers". E.g., A-10s are deployed in Tactical Fighter Wings - there is no such thing as an "Attack Squadron" in USAF. While the A-10 possibly only escaped an F-designation by using its speed ;-), there was no reason (for the USAF) not to call the F-117 a "fighter". After all, there have been fighters with marginal or non-existing air-to-air capabilities in the Air Force before - F-105 and F-111, respectively.
• The "F" prefix was part of the classified designation, directly connected to the number 117. See discussion below.

Check the link for more information.