Cradling their first child, a beaming Jacinda Ardern and partner Clarke Gayford on Sunday greeted media for the first time since arriving at Auckland Hospital on Thursday. She thanked hospital staff for her care.
Ardern became her party's youngest leader and New Zealand's youngest in 150 years after defeating former Prime Minister Bill English in last October's election.
A spokesman said the message Ardern referred to from Prince Harry and Meghan was a "lovely hand-written note" that looked as if it was written by Meghan.
The name also means "snow", which Ms Ardern said seemed like a "good combination" for Matariki, the Maori New Year, and the winter solstice, which was when her daughter born.
The news of the baby's birth was made public on the New Zealand leader's official Instagram account with a photo of the new parents smiling with their daughter, whose name has not been released.
"To me in those moments and those messages, we just saw a reflection of how open and generous New Zealanders are, and that has been the case all the way through", she added before joking: "We're all doing really well".
"We'll do what every other parent does and learn the little nuances of the way that Neve wants to work", she said. "It was pretty special".
Arden is the first elected world leader to take maternity leave, handing over the reigns to Deputy Prime Minister Winston Peters for six weeks, and only the second to have a child while in office - both roles she has spent months downplaying.
Aroha means "love" in Maori, while Mt Te Aroha is a 952-metre mountain in the Kaimai Range, on New Zealand's North Island.
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"I won't forget the look on Jacinda's face when she finally held the baby", Mr Gayford said.
The little baby girl, who is absolutely divine, as a very special and handsome name.
At the end of the video, Ardern calls her partner over to stand in front of the camera.
Ardern delivered her 3.31 kg (7.3 lb) daughter in the country's largest public hospital on Thursday afternoon with her partner, television presenter Clarke Gayford, at her side.
"They congratulated the PM and Clarke and said they were looking forward to seeing them in the fall", the spokesperson said.
Te Aroha is also where the 37-year-old prime minister is from.
Gayford is a radio and TV broadcaster, and the host of the fishing documentary show, "Fish of the Day".
At almost three days old the tiny tot will make her first public appearance at 11am cradled in mother Jacinda Ardern's arms with proud dad Clarke Gayford.