I was actually shorter than 5 feet tall until the summer after 9th grade (I would've been 15 at the time). Some time before that, my parents actually took me to an endocrinologist to see if I could get some growth hormones. I did end up getting them. I'm now about 5'6". Not tall, but not bad, considering my dad is 5'6" and my mom is 5'0".

I'm not sure exactly what they do, but from my experience and the little bit of reading I've done, they're what caused me to undergo puberty (I wonder what things would've been like had I not taken them) and begin growing as well as starting to develop sex characteristics (body hair, lower voice, etc.).
Before I started, they had to X-ray my wrists to see how much room for bone growth I had left, which is, as Dena said, found in a bone gap. Since this gap closes as one ages, there will come a point when one is no longer able to benefit from the growth hormones. They also took blood samples, which caused me to pass out once or twice. That's how I discovered I get queasy when I see my blood being drawn.
If at all possible, I'd urge your nephew to visit an endocrinologist to see if this is an option. If your nephew doesn't do it, and doesn't end up getting a growth spurt, he'd probably spend the rest of his life regretting it. If it turns out that the growth spurt comes naturally, then everything is fine anyways.
One thing to note: I had to receive daily injections of the hormone from my parents. I didn't like shots to begin with, and with my inexperienced parents doing it (I'm thankful they did it, of course), it did sometimes hurt. Plus, I probably had roughly 5% body fat or less, and since the shot was supposed to be injected into body fat, it was sort of a challenge to find a good area to inject it.
Let me know if you have any more questions, and I can try to answer them. I'm by no means an expert in endocrinology, though.