They were joined by an Air Rarotanga aerial search, and a police patrol vessel.
The search was scaled back and eventually called off two days later.
Lapana was raised in South Auckland but had travelled to the islands to see where his family were originally from.
His father Junior Tupou spoke to the Cook Islands News yesterday.
"He didn't want to leave," Junior told the publication.
"He told me New Zealand was too cold, and he wanted to stay in Rakahanga."
Junior said his son was so excited and happy to discover the beach where his family came from.
'"Every day was about fishing, being at the beach, freedom and enjoying being a kid," Junior said.
He said his son was a fan of the All Blacks and was excited for the Rugby World Cup.
"When the All Blacks do the haka, he does the haka too," Junior said.
He told the newspaper that recently he was shopping with his son at a South Auckland mall and the youngster spotted Aaron Smith.
"He recognised him first and called out," Junior said.
"I had to go and ask him for a photo, just for Lapana."
Lapana is the youngest child of Junior and his wife Matapo.
He is survived by siblings Mehau, Danina and Metuakore.
He attended Poetiare o Rongomai preschool in Otara and Chapel Downs School in Flatbush.
Cook Islands police say the death of the boy will be investigated for the Coroner.
The Ministry of Transport there will also investigate his disappearance and presumed drowning.
Back in New Zealand, a Givealittle page has been set up to help Lapana's family.
Monovai Melanie Arai, a cousin of Matapo, set up the page and hoped donations could help Junior and Matapo pay for a funeral for their boy.
"Their hearts and love for their Prince Lapana was pure and true," said the friend.
"We will continue to pray for his return home."
Earlier today she updated the page.
"I want to personally thank everyone who has donated to this cause and truly appreciate all the love from all over," Arai said.
"Although the hopes of Lapana returning are strong as are our prayers, there is comfort in knowing family, friends and the community all over the Pacific have been a great amount of support over these past couple of weeks to the grieving family."
As of this evening more than $1200 had been donated.
If you would like to donate - click here.