According to Grant in a promo video for Horse (which I may have somewhere) I believe he said her name is Rachel Callahan. I recall him introducing the video stating something along the lines of, "starring the very beautiful Rachel Callahan; she is gonna be famous one day".
I tried searching her name a while back but I found no results. My assumption is that she is a relative or family friend of one of the band's family and/or friends (possibly a girlfriend at the time?) of one of the band members.
Although the part about her name is not said in this clip, part of the promo it was in can be seen in the following:http://www.youtube.com/watch?v=wwEWKl1iEX4
On a related note, I've had the following info about the video saved on my computer for some time (don't remember offhand where I found it though):
In mid-1986 I approached Dave Pirner with the concept of shooting a super 8 music video for his band. The plan was to create the video independent of their record label. Dave, already a fan of super 8mm do-it-yourself film making, thought it was a cool idea, so with a few rolls of super 8 film, old cameras, and a gruesome vision, we set about the task of creating this grisly masterpiece. Since it's debut, Artificial Heart has had the distinction of being banned by U.K. television, U.S. MTV, and various VHS compilations. During the late 90's, Matt Pinfield of MTV's 120 Minutes, showed it frequently, boosting this gory cult favorite into the mainstream spotlight. Dave’s mother sewed the surgeon’s costumes, and the operating room scenes were realized through creative use of raw chicken, stage blood, beef liver, and joke-store rubber body parts. This video is dedicated to the memory of Karl Mueller.
(Edit: found the source: http://feeds.revver.com/2.0/mrss/qt/search/8mm