Several people who own condos on Little Cayman will rent them on a short-term basis. Let me run through some of the things which come quickly to mind:
1. The Conch Club condominiums, which are about 1/4 mile east of the Little Cayman Beach Resort, are often available. Since these were developed by the same person who owns LCBR, there is a rental office in the Beach Resort. Contact them for more info.
2. About 200 yards west of Little Cayman Beach Resort is "The Club", also developed by Dan Tibbetts. This was built as a luxury time-share, but the last I heard only one of the eight units had sold. Dan has since started renting the unsold units.
3. There are a couple of small cottages just to the west of Conch Club which are available for short term rental. Contact Cayman Villas for more info.
4. Down by the lighthouse on the western tip of the island there is at least one, and maybe two, condos which are available for rent. Contact Betty Bua at Tranquil Realty (
http://www.tranquilrealty.com/index.html) for more info. Betty would never forgive me for posting her email address publicly, but I will share privately on request.
A roll-it-yourself vacation on Little Cayman isn't as awful as people are trying to make it sound. The trick is to load up on meat at Foster's Food Fair on Grand Cayman and bring it with you. The store on Little Cayman doesn't usually have much more than frozen chicken, frozen sausage, frozen pizza, etc. They do have reasonable produce, though, and you can get the other necessities there. But it is priced somewhere in the range of 2x-4x what you're likely to consider reasonable!
In general, your best value is going to come by staying at one of the resorts. But each year we have plenty of people who rent cottages or condos because they want more privacy. I've thought of doing this with my own home there, since I'm always off the island from May-September, but there are just too many hassles involved in doing that.