Marurai was prime minister for six years, from 2004 to 2010.
He had been in parliament from 1994 until his resignation, because of ill health, in 2017.
Marurai was born in July 1947 on Mangaia and had been described as of that island's favourite sons.
Like a number of the Cook Island leaders, he was a product of Otago university, and became a teacher before entering politics.
The current prime minister, Mark Brown, expressed his condolences and sympathy on behalf of the people and government of the Cook Islands.