Was the intended purpose of the F-117 Nighthawk originally a stealth-fighter?
The short of it is that the "A" designation for a ground-attack aircraft has historically been a Navy distinction. The USAF/USAAF/USAAC has not traditionally differentiated between an aircraft designed for tactical ground strikes versus a dogfighter/interceptor. Most WWII-era Air Force planes were capable of both, and would be called "multirole fighters" in today's parlance (including the famous P-38, P-40, P-47 and P-51). Between WWII and the adoption of the 1962 Tri-Service Aircraft Designation System, the USAF used "F" almost exclusively for any one- or two-man aircraft performing either the air-to-air or tactical air-to-ground role (using "B" for aircraft intended for larger-scale strategic bombing). Thus, the "Century Series" aircraft, many of which were primary ground attack aircraft, all got the "Fighter" designation even when their mission profile was exclusively air-to-ground, such as the F-105 and F-111.

The Tri-Service Aircraft Designation System was patterned most closely to the Air Force's pre-existing system, but resetting all the counters (the Navy's system involved much lower numbers, many of which were grandfathered into the new system) and adding mission designations common in the other branches, including the Navy's Attack designation. The Air Force has more or less ignored many of the finer points of this system when designating its own aircraft; the A-10, and its A-X competitor the YA-9, are the sole airframes designed primarily for the USAF to carry the Attack designation (which is fitting as both planes were designed from the ground up to kill tanks and other armor). The other A-designated aircraft to enter service with the Air Force were Navy planes first (including the A-1 Skyraider and A-7 Corsair II). 

The F-15E should, by tri-service rules, be designated AF-15E (emphasizing its modified mission of tactical strikes over its air-superiority pedigree). "F/A" as used for the Hornet is also non-compliant, and stems from an original desire to have separate fighter and attack variants of the Hornet which ended up being rolled into one plane. "FA" or "AF" would suffice not only for the Hornet, but also the F-16 Falcon and the F-35 Lightning II (yet another aberration, this time in the numbering, keeping the number from the X-35 tech demonstrator instead of being the F-24 as expected).

The F-117 is a very special case. The airframe received its designation after 1962 and so should have been designated either A-14 (as its mission has always been tactical ground strikes) or F-19 (keeping with the USAF's tendency to label any light offensive aircraft a "fighter"). Development of the F-117 from the Have Blue project was an open secret, and it is widely believed that the Nighthawk is the missing "F-19" in the U.S. numbering system (the F-20, a variant of the F-5 powered by a single larger turbofan, having been announced and demonstrated long before the F-117's existence was officially acknowledged), with the pre-1962 designation intended as misinformation.