If your cousin doesn't show any interest in anything then don't include him in anything. Seriously, he will just have to learn by himself. If you include him I feel that you will regret it as he will be nothing but a burden. There are plenty of people that would love to be in a position to travel across the US so if he doesn't appreciate the chance...
I don't mean any disrespect, but your counsin sounds like he needs more than just motivation, but a swift kick in the ass, along with a good dose of reality check. I don't see why he *deserves* a free trip anywhere. I like the idea of charity work to see how lucky he is. Or, if he's dreamer, encourage him to go on job interviews. He will get a better clue after a few rejections on what skills/knowledge each job requires. I wouldn't go on any trip with him, I think it'd just be a waste of your time and money.
Let your cousin be who he wants to be and stop trying to turn him into what society believes to be an ideal citizen living the ideal lifestyle. It's his trip, do what he wants to do. If you took me hiking on my trip I'd kick your ass.
I looked into it for myself (and my brother who is hopeless like this fellow I don't even know if he'd try at gaming which is the one thing he has interest in!) anyway usually it doesnt work out going to those schools better off learning and breaking in on your own. Why the hell you wouldn't want to go road tripping I don't understand but like I said my kid bro is like that all he does is game. I do to but when it comes time for a nice road trip I'm going. Just finished one in fact which was a blast!