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.

They began by announcing the baby would be called Neve Te Aroha Ardern Gayford - or Neve Gayford for short.

"When we met her we thought she looked like she suited the name," the 37-year-old prime minister told reporters.

"Also it means, in various forms, bright and radiant and snow, which seemed like a good combination for Matariki (Maori new year) and for solstice."

Aroha in Maori means love and Te Aroha is also the name of a mountain near where Ms Ardern's family come from.

"Te Aroha was our way of reflecting the amount of love this baby has been shown before she arrived and all of the names we were gifted along the way (by various iwi - or tribes)," Ms Ardern said.

She and Mr Gayford had opted to spell the name Neve - as opposed to Niamh, for example - for simplicity, with Ms Ardern joking about the unusual spellings of Clarke and Jacinda.

Earlier this year they said it was likely the baby would take her father's surname.

The couple on Sunday described to reporters the first moment they met Neve.

"I won't forget the look on Jacinda's face when she finally held the baby. It was all a bit of a blur for the both of us ... She looked absolutely just stunned and very, very happy," Mr Gayford said.

Since the birth on Thursday, Ms Ardern has been inundated with messages and gifts from world leaders, including the Queen, as well as the Duke and Duchess of Sussex - who have sent a handwritten note saying they look forward to meeting later in the year.

"But as equally special to us were just those people who took time to send a little note, or a blanket or a set of booties," Ms Ardern said.

The family will now leave the hospital to spend a stint at home and away from the spotlight.

"Over the next six weeks we'll do what every other parent does and learn the little nuances ... and just figure things out as we go."

Ms Ardern has become the first elected world leader to take maternity leave but will still be on call for major government issues.

Mr Gayford, a 40-year-old television presenter, will be the child's primary carer once the prime minister returns to parliament.

"Clarke's being as much or a role model here as I am," Ms Ardern said.

"I hope for little girls and boys there's a future where they can make choices about how they raise their family and what kind of career they have based on what they want."

Australian Associated Press