I investigated GH a couple of years ago for DS1. He was borderline for the opposite, and the GP was wondering if he had precocious puberty as his height had skyrocketed and he was suggesting daily hormones to slow him down.
So I spent nights reading a lot about what affects/activates/slows GH and went privately to see an endocrinologist.The endocrinologist told me that growth is determined by your genes but lifestyle has an up to 20% influence. You can either not reach your height or go well beyond.
GH is released during sleep, so the more it has, the higher the levels. That's why children grow more during holiday because they sleep more. Gluten is a GH inhibitor which can in part be compensated by large amount of milk. High quality and quantity of protein are hugely important for growth. Another important factor is extremely high intensity sport, such as a couple of sprints at maximum level, far more effective than a marathon. The intensity should be such that you can barely speak after it. High intensity activity releases a rush of GH.
So to sum it up, diet, sleep and sport.
This is of course for the general public. I am not talking here about children having syndromes like the previous poster, deficiencies caused by illnesses or traumas.
Nowadays, GH is made synthetically, so no more of the terrible effects and deaths when it was taken from bodies usually from old people who had died in homes.
I would go for it. From what I read years ago, it is dafe and life changing for the children with growth problems. What is you DS's problem OP?