Consider that maybe you're asking the wrong question. Maybe you ought to be asking why exactly is he not motivated to go out more? Is he also not outgoing when he's at the other parent's house? This motivation must grow in him naturally, not really be stuffed into him.
I have been an avid gamer myself when I was his age (18 years ago), and I spent much time at home just playing videogames whenever none of my friends came out for some reason. Essentially, I would just not go out if I had no reason to.
That I didn't have a girlfriend was an important factor. I wanted one really badly, but I didn't see it happening in the foreseeable future, so I just spent my time doing what I felt was the next best thing. And even though I loved playing videogames that much, a girlfriend would've easily made me forget they existed.
Summer youth camps and activities always seemed overly boring to me, probably because I was raised in the woods and I had already lived through much more appealing adventures and activities on my own.
My best guess is that this is a temporary phase, in which he may be needing something refreshing in his life, and as soon as he has a chance to get a job he'll meet new people, see new places, feel the freedom to wonder off on his own, and it will all feel quite refreshing. He'll also meet new girls, and most likely fall in love, or at least have some fun.
If my prediction isn't correct, then maybe he has a problem of some sort that he doesn't know how to overcome. I had complexes of my body, and while that may not sound serious, it was what made me not live through the most important stuff I should've lived in my teen years... How to deal with a boy's problem of this sort is something I unfortunately don't know. I was shy and introverted when it came to mine. The only way I was ever able to let it out was thanks to a close (girl)friend who noticed my problem, decided to try to help me, and told me she had had it too, which made me feel at ease. I was never able to speak to my parents or anyone else about it.
On a side note about videogames: before I had a PC and many games to play and apps to fiddle with, I was constantly sighing and complaining that I had nothing to do. From then on I had something entertaining to spend my spare time with. It's not a bad thing, unless it's the only thing we do in life from dawn till dusk. If it looks like an addiction then there is probably something hindering or demotivating him (remember, an addiction isn't the actual problem, it is only caused by the actual problem).
That said, if that's not really the case, then videogames are actually not a bad thing. I learned a few valuable things with playing videogames. Namely that cheating removes the satisfaction and sense of accomplishment from achieving things (a notion I still apply to all of my life), and that greed never pays any better than perseverance and patience. I also keep some nice nostalgia from those times.
I hope I gave you enough food for thought, and I hope it turns out to just be a phase. The sense of freedom after being 18 and having their own money and no questions asked, will probably change things.