This would be the early 80’s, when music videos first started playing on TV. I don’t remember the song or the band, but the story of the video was of a teenage girl who has a crappy home life and runs away. At the end of the video she finds this remote cottage where the band is and at the very end she’s sitting happily in the cottage with the band while they jam. I think the word “rebel” or “runaway” might be in either the song title or the lyrics.
Does anybody remember this video?
Nope, it was earlier than Bon Jovi and not as metal-y.
I keep thinking the band’s name started with “M” and they were a one-hit wonder.
Nope. Probably a bit older. (Have you seen the “literal translation” of this video? It’s hilarious.)
Is the singer male or female? Does the plot of the video go along with the lyrics of the song?
It was a band of just guys, and yes I think the plot of the video goes along with the lyrics. Something about a girl running away or rebelling. It was rock 'n roll but more pop-like, not heavy metal. All I can picture in my head is the final scene where the girl is in the cabin with the band and the camera pans away. The girl was your typical 80’s teenager, big hair (dark hair, IIRC) and either a leather jacket or a jean jacket.
I’m starting to wonder if I dreamed this video because I’m usually really good with 80’s music. But I remember seeing the video a number of times and wishing I looked like the girl in it.
“I Will Follow”, by U2? (“If you run away, run away, I will follow”) Probably not, hardly one-hit wonders. Hmm, “Hot Child In The City” was 70s. “Rebel Rebel,” David Bowie, was also 70s I think.
In “I Want You Around,” by the Ramones, the band actually ends up at HER house. LOL, Dee Dee Ramone she finds in her shower. But they do all have leather jackets on. 
Gimme some time; I’m working on it.
Hey, lookee here:
songs with “run away” in the lyrics.
scroll down a little past the artists and albums with those words, down to “songs.”
They are Not ordered chronologically, and it looks like there are pages and pages and pages.
And here’s songs with “rebel” in the lyrics.
BTW, if your browser has a “find” function (under the edit menu in windows) then you can tell it to find 198, and it will automatically take you to each song from nineteen eighty-whatever. Press the down arrow to go to the next one.
Thank you brujaja, that gives me lots to work with. 
I’m sure this isn’t it, since it doesn’t fit the ending at all, but the idea of a teenage girl running away from a bad homelife immediately made me think of Love is a Battlefield by Pat Benatar.
The relevant scene comes in around the 40 second mark. I post this just in case you may be conflating it with a different video.
Well, when I read the OP I thought it sounded familiar. I picture the runaway girl being blond; and wearing some yellow / pink combination. Then I realize that doesn’t even narrow down the 80s video to any discernible degree!
I also want to thank the OP for the trip down memory lane; because I went to youtube and found the billboard top songs of the 80s. I’m listening to Billy Joel right now; next up: Blondie - Call Me. Woo Hoo. I hope you figure out your song!
Wrong decade, but how about Runaway Train by Soul Asylum. The band is sitting in a cottage ‘sort of’ and there’s lots of people running away (kinda).
That’s the first song that came to my mind, until I read the whole post. That is a really dark song and video all the way through. Nothing remotely happy in it at all.
C’mon, man - you just can’t post random songs that have the word “runaway” in them. You’ve got to put some thought into it.
This isn’t the Thread Game forum.
Who said anything about it having to be a happy video?
ETA, I see the OP said that she’s ‘sitting happily’ but that doesn’t mean it was a happy video.
Could it be Rod Stewart’s Young Turks?
It has the runaway theme, but the band isn't jamming at a cottage but in some rundown urban setting.You know, the very first thing I thought of was “No Rain” by Blind Melon. One-hit wonders, “M” in the name, and the story of escaping an unfriendly world to a lovely place in Nature with like-minded people is the same, although depicted very differently from the scenario Soylent Juicy has in mind.
Skip the tap dancing if you must, it lasts about 30 seconds; but it is integral to the story for anyone who’s never seen this.
I love this song. :)