The Northern California band was led by lead singer Tommy Heath and guitarist Jim Keller. The group was originally called Tommy and the Tu-tones; this eventually was shortened to Tommy Tutone. While there is some mythology around the song, it was simply a made-up story as there was no Jenny. The song was co-written by Jim Keller (Tommy Tutone) and Alex Call (who also wrote "Perfect World" for Huey Lewis and the News and "Little Too Late" for Pat Benatar).
As alluded to before, Tommy Tutone had a minor hit in 1980 with "Angel Say No," which was my first choice for the blog. However, it does not appear that a video was ever made for that song, so we put up an audio clip on the ERV Facebook page. "Angel Say No," from Tommy Tutone's 1980 self-titled debut album, peaked at #38.
Unfortunately, the band's third album, 1983's National Emotion, barely broke the top 200 album charts and the group broke up. Jim Keller remained in the industry, while Tommy Health apparently left the music business for a time. They both have occasionally recorded and performed, and I believe that Tommy Heath has a version of Tommy Tutone together as of this writing.
Cool trivia fact: The actress in "867-5390/Jenny" is Karen Morton, Playboy's Playmate of the month for July 1978. For those who are interested, there is an interview with her on the Noblemania blog.