I had a place at Hilton Head until a few years ago.
In my mind Rose Hill is a dog track even though it was recently redone. I think the layout is very poor, it is cheap however. The great tracts at Hilton Head are expensive. If you want decent at mid prices you might try Hilton Head National, The Golden Bear at Indigo Run, Eagles Landing (think that is right) there is also another next to it, one is a Palmer and the other is a Love course. Palmetto Hall has several decent courses as well.
You can get some afternoon discounts at some courses and a few have a deal where your son if under 18 plays free. When you get there grab a golf guide, or go online and download one, it's free. Has all the local courses in it and some discounts.