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.
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.
"For a matter of months we kept a short list of names and we wanted to wait until the baby arrived to really see which one felt like it really worked", she said.
The couple's first child was born on Thursday at 4.45pm, weighing in at 3.31 kgs.
She said, like most parents, they went through a struggle of trying to decide on the name, but settled on Neve Te Aroha Ardern Gayford, with Ardern acting as a middle name and Gayford the surname.
"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. Can you remember leaving hospital after giving birth?
When we met her, we thought she looked like she suited the name.
The First Couple looked radiant and couldn't wipe the smiles off their faces today as they made their first public appearance with newborn daughter Neve Te Aroha Ardern Gayford.
"Of course we need to make sure that in all of the things we do, whether it's a pay round or a health policy, we reflect the value of that workforce", she said.
She added: "I hope for little girls and boys that there's a future where they can make choices about how they raise their family and what kind of career they have that are based on what they want and what makes them happy".
The pregnancy news late a year ago came as a surprise to Ms Ardern, who then had to simultaneously deal with both intense negotiations to form a government and morning sickness.
At the time, she said she'd found out about the pregnancy just one week before her Labour Party formed government and that it came as a surprise to her, and her partner, as they believed they would not be able to conceive naturally.
Jacinda Ardern is New Zealand's first leader to give birth while in power.Ardern insists it will be business as usual during her six weeks off. Thank you so much for your best wishes and your kindness.
The Embassy of the Kingdom of Saudi Arabia also sent a bouquet so large it could not fit in the Prime Minister's room in the maternity ward of Auckland City Hospital.