Unfortunately "Hilton Head" and "cheap" are not usually used together.

BUT that doesn't mean that you can't go there and have a great time!
You have the right idea to start with -- using points and preparing your own meals. Once you get beyond that, however, it
can get costly, if you let it.
The Disney snack bars are not inexpensive. You can limit the number of meals you have there and instead go back to the room or bring a cooler with lunch to the beach house.
Restaurants can range from reasonable to costly. A lot of the restaurants offer early bird specials which allow you to eat less expensively AND avoid the long waits that are common during the busy months.
Groceries on the island can be expensive as well. Prices will likely be higher than you are used to paying at home, just because of the location. You can help that situation by bringing a lot of the staples with you (condiments, salad dressings, staples, etc.) instead of buying them there. There is a Sam's Club not far from the Disney resort, if that helps you, or you can do a major grocery "run" to an off-island grocery store. (Personally, I would rather pay the higher price and get to stay on the island than to take up my valuable Hilton Head time going to buy groceries

)
As far as activities, you can bring bikes or rent them fairly inexpensively -- the resort offers a "length of stay" option. Many of the nature-oriented activities offered by the resort are free of charge. You can also check out what is offered by the Coastal Discovery Museum for a small donation or free. It's more the food-oriented and kid-oriented activities at the resort that have a price tag attached. Also, there are all kinds of temptations on the island, like golf (regular and miniature), arcades, etc. that can empty your wallet.
A lot of people bring their own beach chairs and umbrellas, which is a BIG savings over renting them at the beach. You can get them pretty inexpensively at the
WalMart, too. Same with boogie boards and other beach supplies.
Hope this helps.
