I first heard this term in the 90s. It seemed like producers couldn’t be bothered anymore with the logistics of a weekly show.
My exposure was also via the Simpsons DVD commentaries, or maybe general articles about the show, sometime around 15 years ago.
Actually, per Wiki, it developed because the term executive producer was being applied to a wide variety of roles.
I think that’s when I first came across the term, too. Except that it was in the context of RTD explaining, a little self-consciously, why he’d adopted the American term “show runner” which hadn’t been used in British TV up to that time (it’s comparatively rare for writers to also be producers in the UK). The specific comparison he used, if I remember right, was Joss Whedon on Buffy.
I also first heard it in regard to David Chase and The Sopranos. Probably eight or nine years ago. Have seen it a number of other times since.
ETA a couple of descriptions from Wiki:
In the United States television industry, a showrunner is a person who is responsible for the day-to-day operation of a television series—although usually such persons are credited as executive producers.
Traditionally, the executive producer of a television program was the chief executive, responsible for the show’s production. Over time, the title of executive producer became applied to a wider range of roles, from those responsible for arranging financing to an honorific without any management duties.
Pardon the redundancy. I missed that Colibri had already posted info from Wiki.
The first time I remember hearing it was also in reference to Russell T. Davies, I might have heard it before then for some HBO shows, I’m not sure.
There does seem to have been something of a sea change in Britain in that regard. If you read about old-school Doctor Who, you will often run into mention of how there were strict rules against the script editor “commissioning himself”–that is, actually writing a script for the show. If Terrance Dicks or Robert Holmes wanted to write for the show while also having executive responsibilities on the show, there were apparently all sorts of hoops they had to jump through, often involving the use of psuedonyms, temporarily stepping down from their production position, etc. The current situation, where someone like Davies or Stephen Moffatt is a producer and also a regular script writer, apparently would have been completely out of bounds in the old days.
Show runner is a relatively new term, as opposed to something like “producer” or “director.” I still don’t think it’s a term you’ll see in the credits anywhere. I get the impression it’s something of an insider term, and this being the age of the Internet, it’s crept out into the common vernacular. Certainly there have always been people who were in overall charge of a particular TV show. People like Carl Reiner for The Dick Van Dyke Show, or Garry Marshall for Happy Days, to name a couple of obvious examples from the past. In those days they were just called producers, but would most likely be called showrunners today.
I think I first saw the term in reference to Brannon Braga and Gene Roddenberry’s other successors in the various Star Trek shows.
It was definitely not used 20 years ago, though the people who would be show runners today (the creators) were clear.
Gene Roddenberry is the obvious example of a showrunner from the 60s. He had the title “Executive Producer”, which was the style at the time. As has been mentioned, the problem was that producers and executive producers magically encrusted themselves onto projects, as the guys with the money wanted to see their names in the credits. And so how do you tell the difference between an executive producer whose contribution was putting up two million dollars and settling back on his poolside lounge chair, and an executive producer who hires the casting director and has ultimate say on casting, chooses the directors and tells them what to do, chooses the writers and has ultimate say on the scripts, chooses and guides the designers who create the sets and props, and on and on. And so the term “showrunner” is born inside the industry and then makes its way into general use. You still won’t see anyone listed as “showrunner” in the credits. They might get credited with “created by” or “executive producer” or “producer” or whatever.
This Slate article says Variety started using the term in 1992. It also says:
He may have still had the title, but he pretty much sat out the entire horrible third season, once he could see the show was going to be canceled. Freddie Freiberger and Arthur Singer (“Tell me again what this transporter thing does?”) were the ones running the show then.
Yes! I miss the days when EVERYTHING wasn’t a freaking soap opera. Gah, I am so tired of this soap opera fad, and really look forward to that going away.
There’s a good documentary about show runners, called Showrunners. It’s a pretty interesting look at some of the heavies in the field (a couple of years old, but still relevant).
Here’s the website and there’s a book too.
Apparently this documentary is unable to be viewed in the US.
That was after the show I was thinking of.
Okay… 
I’m sure they all still retain the title “Executive Producer”; for example, Matthew Weiner is an Executive Producer on “Mad Men” along with 4 other people. Saying that he’s the “Showrunner” is the way to describe him as the guy with the ultimate control within the show’s production.
I get the impression it’s something that’s sort of grown organically with the rise of scripted multi episode and multi season plots; while I’m sure “Three’s Company” had a EP who was the showrunner, his prominence was probably more administrative, as the show was almost completely episodic. There wasn’t much need for someone in the first season to make sure that the writers weren’t writing something for the Furleys, Janet, Jack or Chrissy to do something that would break the plot for the 3rd season.
Contrast that with something like a Babylon 5, Mad Men, Battlestar Galactica (new one), or The Sopranos, all of which had various writers and directors for their episodes, but that needed one person to sort of drive the boat, so to speak, and make sure that everything remained in continuity and didn’t break the downstream story arc and things of that nature.
So with that role being expanded, the term sort of probably cropped up, and is now being applied to whoever the controlling EP is for that show.