Mom2Boys: Getting tee times on short notice isn't hard for an afternoon round, but can be a little tougher for morning tee times. It also depends upon the courses you would like to play on.
As for rates it depends upon the type and location of the course. The lowest you could get would be Rose Hill, in the ballpark of $65. This course is 3-4 miles off the island, which is about a 1/2 hour trip from the center of HHI, and the condition is well-below par. On the "high end" you could play Harbourtown at about $175 -- if you can get a tee time.
Some of the courses I liked.
(Middle of the Road Pricing - $75-$100, depending on time of day)
-Old South - just off HHI on the bay
-Hilton Head National - just off HHI; typical Gary Player design
-Palmetto Hall - has two 18-hole courses (I prefer the "Cup" course)
Hope you find this helpful.