I'd love to try to help with your question... but unfortunately, there may not be an easy solution to this.
First, it sounds like your son's diagnosis is idiopathic short stature - is that what you've been told? Essentially, that just means that a person is short without a medical reason (like kidney disease or a hormone deficiency). Usually, kids with idiopathic short stature are shorter children because their parents are shorter adults - so it's sort of "in the cards" for them, genetically speaking.
For children with severe idiopathic short stature, many endocrinologists will prescribe recombinant human growth hormone injections, as you suggest.
These injections, though, have a downside. For one, although they're effective, they're only modestly effective for children with idiopathic short stature (compared to children with growth hormone deficiency, for example). Basically, at your son's age, you could probably hope for an improvement of only a few inches versus his height without growth hormone. Another disadvantage is that the medication must be administered as a daily injection. I emphasize daily, because if you only give the medication a few days a week, the body tends to not make growth hormone on the days you don't give it - so to attain a real improvement in height takes giving the medication every single day for a sustained period of time. The injections are not especially painful, but the fact that the medication must be administered as a "shot" can be a significant hurdle for some children and families. And finally, the medication is expensive.
Generally speaking, insurance companies will only cover growth hormone for children with idiopathic short stature whose height percentiles are much lower than your son's - usually less than 3rd percentile. That means that you'll end up having to pay for the medication out-of-pocket...
To give you an idea about how expensive that might be, I just looked up the cost of Genotropin. A 5 mg cartrige for injection costs $381.99... and I would ballpark that your son would need at least one of those every 4-5 days. There are other brands on the market, so you may be able to get the medication more cheaply... but I bring it up for the sake of example to give you a sense of just how expensive it might be unless your insurance covers it.
If the cost is not a concern for your family, then I think the best advice I can give you would be to obtain a second opinion from another pediatric endocrinologist. Unfortunately, I practice on the East Coast, so I can't recommend particular physicians - but your best bet would be to look at any children's hospitals or medical schools in your area (or surrounding areas). Another endocrinologist may be willing to prescribe the medication after discussing the risks and benefits with you.
I know this was not the answer that you were hoping to hear, but I hope that this information is helpful to you and helps to put this issue into perspective.
Best wishes to you and your son...