A weak positive result could indicate a recent infection or a historical case.
The 10-year-old child was travelling with his mother, who is fully vaccinated, and two siblings. The three children are not yet eligible to be vaccinated.
The family are in isolation at the Edgewater Resort, a MIQ facility on Rarotonga, but so far neither the positive case nor any family members have shown symptoms of the virus.
Brown said they will remain in the facility until there is no risk of the virus spreading into the wider community.
All border workers who met the flight would also be tested, as an added precaution.
The Prime Minister said they had been preparing for the time to reopen their borders and catching this case at the border had shown the value of their preparations.
"I want to reiterate that for many months we have been planning for how we handle the border reopening and dealing with any potential coronavirus threats, I further reiterate the importance for all eligible members of our population to be vaccinated, to give us that added layer of protection against serious illness and possible death."
He said the vaccine was available to all eligible people and would be available to 5- to 11-year-olds once given the all-clear for use in children.
"The ability of this virus to cross borders can be seen worldwide, and that is why border workers wear Personal Protective Equipment [PPE] and follow infection control procedures at work – to eliminate the risk of catching the virus through their work.
"I hope you will join me in wishing the family a speedy recovery."