Rarotonga, and to a lesser extent, Aitutaki, have a good choice of accommodation, from the luxurious to the very basic. As you might expect, on the other islands, tourist accommodation is very limited. This is a non-commercial web site, so this page is not intended to provide all the answers to the question of where to stay. There are plenty of web sites with information, and links to some of them are provided below.
If you're after a privately owned, luxury home in a superb location, I would strongly recommend my friend Karen Johnson's place, Taakoka on Muri Beach. You'll get a very warm welcome. Tell her I sent you! You can book by contacting Karen direct on malsam@oyster.net.ck. She's also a very talented fashion designer and her garments can be seen not only in the shops but also on the staff in some of the best restaurants and hotels. She'll make to order for you if you give her enough notice.
Karen and her husband Mal are pictured below, along with some pictures of Taakoka. Click on the thumbnails for a larger picture.