The Cook Islands museum is attached to the Library and you pay here first to enter. Opening hours same as the library.
Its not particularly big or organized. Just a large room filled with various bits and pieces. The exhibition includes a variety of items - types of sealife, mainly shellfish found here, the old treaty documents, European pictures, and accounts of the islanders. It's all a hodge podge really, but the museum also displays several large pieces of sculpture by local artists and some of these are rather lovely.
The museum does give you a bit of insight into the Cook Islands and at $2 doesn't exactly break the bank. I would recommend a visit, probably on a rain day though.
NB visitors can join the library for $25 ($15 deposit).