The newly minted millionaire was reported to have just returned from a holiday with his family when they were told the news.
"My husband was literally pumping the air when he was on the phone," said the unnamed child's mother, according to local media.
It is not known how the winner reacted.
Bonus Bonds are an investment that pays out thousands of prizes to savers, lottery-style, every month. NZ$1m is the biggest monthly prize.
The child's Bonus Bonds account was reportedly set up when he was born with just NZ$250 ($180; £140) worth of bonds, given by a relative.
ANZ bank, which runs the scheme, confirmed the win but said it was respecting the family's wishes by not giving further information about the winner.

Oxford City Council said the money had mostly been used for "ground investigations of possible sites" but nowhere suitable had been found.
Two cemeteries still have space, in Wolvercote and Botley, but they are expected to be full by 2018 and 2021.
The council said it had not given up and was "still exploring options".
Linda Smith, board member for leisure, parks and sport, said the council has been "searching for a suitable new burial site for many years".
She added: "But ultimately, as with new housing sites, we have run out of suitable land within Oxford.
"So far all the council-owned sites that we have identified have, following ground investigations and surveys, had to be discounted.
"Either due to the size of the site, the ground conditions, a high water table or a covenant restricting the use of the site."
After the two remaining cemeteries are full the council said only the reopening of family plots, the use of a few reserved plots, and the interment of ashes would be possible.
The last increase in burial space in Oxford was in 1932.
A New Zealand toddler has won NZ$1m ($726,600; £560,700) in Bonus Bonds prize money - the youngest ever winner of the investment lottery scheme.
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A council has spent £30,500 unsuccessfully searching for new burial sites in Oxford over the last four years.