A committee with 19 members was convened in January by Pa Marie Ariki, a member of a group of tribal leaders that advises the government on matters related to culture and custom, Mr. Mataroa said. So far, the committee has gathered 60 names from the public and has the support of tribal leaders across the Cook Islands.

Mr. Mataroa said the committee was considering names with these requirements: It should not be more than 15 letters, have no more than four syllables and be easy to explain in English. Mr. Mataroa said the group hoped to narrow down the list by April.

Names would be presented to the government and could be voted upon in a referendum, Thomas Wynne, a government media adviser who is on the advisory committee, said Tuesday. If approved by the public and the government, an amendment to the constitution would be needed to change the name.

Deputy Prime Minister Mark Brown, a member of the Cook Islands Party, did not respond to a request for comment Tuesday. But Mr. Brown told Radio New Zealand in a March interview that he supported the name change and wanted to hear more comments from Cook Islanders.

“I’m quite happy to look at a traditional name for our country which reflects the true Polynesian nature of our island nation,” he said. He added that it seemed likely that a referendum would be needed to make the change.

The Democratic Party leader, Tina Browne, did not respond to a request for comment on Tuesday.

Mr. Mataroa said he and his fellow committee members were seeking public opinion, a challenge given that the population is spread across more than a dozen islands and two million square miles. “It is going to be a long, expensive process,” he said.