I am 40 now, but sympathize a lot with your friend, as 16 was the age when I first gained a lot of weight (and it was a REALLY big amount) after having been normal weight up to that point. It was hard, hard, hard being an overweight teenager. It's already an age when you really feel everyone is scrutinizing your every move, so getting up the courage to go work out in front of people when you KNOW you have some weight to lose takes a lot of strength. And as we all know, it takes a lot to get yourself going anyway -- she's just facing that extra mental obstacle. (Even if she's acting as if it's just laziness, I would bet that at least part of it is just sheer nervousness. )
As a practical matter, I think getting yourself into the habit of working out is the hardest thing to do, no matter what age. Your body fights you, and you just feel like there's such a long way to go. But if you can get there, and just DO it often enough, everything falls into place before you know it. See if you can get her to commit to taking it just ONE workout at a time (and then another, and another). Even better if you can find a class (or some other activity) that she actually likes. Another thought might be for YOU to ask HER to help you. Say you really need her to hold you to the workouts, so she's actually feeling responsible for more than just herself. (You may have already tried this, but worth a shot.)
Good for you, by the way, for trying to give her a hand. Really, you're already doing one of the best things you can do for her by letting her know you care, and want to help. (And that you're her friend not matter what.) No matter what she does in the near term, the very fact that she knows she has a support system will help her reach the point where she is ready to really commit to herself and her health. I was in her shoes once, and didn't get a positive word from anyone. I really could have used a friend like you.
Starting Weight: 260 lbs. (November 2010)
Quasi-Maintenance: 165 lbs. (June 2011)
Goal Weight: 150 lbs. (reached December 8, 2011)
Size 22W to Size 4
Currently maintaining within 5 lbs. (+ or -) of goal. I weigh daily, but also use a tape measure and the fit of my clothes to guage how well I'm doing.