She said the couple struggled for months trying to choose a name and made a decision to wait until the baby was born to figure out which one fitted best.
Te Aroha is a rural town in the Waikato region of New Zealand's North Island, and is also the name of a Maori song about love and peace.
"When we met her we thought she looked like she suited the name", she said.
Ardern answered questions with a broad smile while rocking Neve back and forth as she slept.
Te Aroha was to encompass all the names suggested by Maori iwi and also reflected where Ardern had grown up.
"We're not placing any great expectations on this baby - except happiness and love", she said.
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The dad looks pissed , and people can be heard remarking "no no no", "sir, he is scared" and "that's not allowed". Suddenly, the priest appears unable to take it again and deals the hollering baby a quick slap in the face.
Ardern became New Zealand's youngest prime minister when she took office through a coalition deal previous year after an inconclusive election, and now became the first woman in the country's history to give birth while in office.
The Prime Minister's office (PMO) says the trio will depart Auckland Hospital shortly after.
"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".
Laurell Ardern said she'd had people come up to her saying that her daughter had been an inspiration, including one woman who anxious a job was too hard for her, but then thought 'Well, if Jacinda can do it, I will do it'.
Ardern's office said Saturday she plans to make her first public appearance since the birth when she leaves Auckland City Hospital on Sunday morning.
In the video, Ardern takes a few minutes to tell the story behind their daughter's name, Neve Te Aroha Ardern Gayford, as well as to thank staff at the hospital.
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As expected, that's behind the $18.5 million earned in previews by Jurassic World (2015) but is still a good number on its own. The Hollywood Reporter broke the news about Jurassic World: Fallen Kingdom's Thursday box office tally.
"Baby slept a lot more last night ... but the PM admits she didn't sleep much while baby slept which is probably because she is still very excited about everything". "So, I'm dying to see it and hold her and just see what's she's like".
All the presents for baby Neve would be passed on to another family once they had been used, Ms Ardern said. Deputy Prime Minister Winston Peters has taken over as the acting prime minister.
Ardern has yet to announce a name for her daughter, after earlier saying the couple was having difficulty deciding.
"But it's equally special to us were just those people who took time to send a little note, or a blanket or a set of booties".
Mr Gayford, a 40-year-old television presenter, will be the child's primary carer once Ms Ardern returns to parliament.
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That means the victor would be in good shape to advance past the group stage, and a draw could end up benefiting either or both. And the 26-year-old admits he's aiming to replicate his efforts with Liverpool with his country this summer.