"My youngest, Hughey, is just 6 months old so it's going to be special having them grow up together."
Fraser said Ardern and Gayford had played an amazing role of uncle and aunt to her children.
"So we look forward to repaying the favour and, of course, meeting the little one soon. There is no doubt they will be amazing parents," she said.
Ardern's big sister, Louise Ardern, was also excited to welcome a new playmate for her daughter.
Louise Ardern famously gave birth to a baby boy while her sister was being sworn in as the 40th Prime Minister of New Zealand.
Welcome to our village wee one.
Feeling very lucky to have a healthy baby girl that arrived at 4.45pm weighing 3.31kg...
Posted by Jacinda Ardern on Wednesday, 20 June 2018
"I'm so excited to be an Aunty and can't wait to have cuddles. Based on what I've heard so far she seems very alert - just like my daughter Isabella who is very excited to have a baby cousin," she said.
Ardern's father Ross and mother Laurell said they were "very proud" of the new addition to their family - their third grandchild.
"We're a family that are fairly well grounded and to see this on the international media is absolutely different - I can say that without a shadow of a doubt," Ross Ardern told RNZ.
"But this is news that Jacinda wanted to share with New Zealand, and for it to be part of New Zealand, and I think her and Clarke have managed to do just that."
Ross Ardern said the baby girl would be good company for their other two grandchildren. He was part of the support team "as all members of our family will be".
Ardern's aunt, Marie Ardern was "absolutely over the moon" by the birth.
"Our Jacinda and Clarke have a wee girl and she is beautiful," she said.
Ardern and Clarke announced the birth on social media at 6.15pm.
"Welcome to our village wee one," they wrote. "Feeling very lucky to have a healthy baby girl that arrived at 4.45pm weighing 3.31kg (7.3lb).
"Thank you so much for your best wishes and your kindness. We're all doing really well thanks to the wonderful team at Auckland City Hospital."
The little girl's name is yet to be announced, but nurses at the hospital said she was "very alert and one hungry baby".
This morning it was announced that the PM will remain in hospital for a second night and no further public events were planned today.
Her baby was one of 24 born yesterday at Auckland hospital.