In a Monday phone chat with herTodayshow co-anchors during herfirst official day on maternity leave, new mom of threeJenna Bush Hageropened up about how her father is reacting to the latest addition to the family.

“My dad was a little mad that the name wasn’t George but [was] just kidding,” said Bush Hager, 37, who welcomed her son with husbandHenry Hageron Friday. “He said [Henry] was a cool name.”

In fact, their new sibling’s nickname is all his sisters’ doing! “We weren’t exactly sure about the name, but then his big sisters arrived and they called him Hal Pal,” Bush Hager recalled. “So we thought, ‘This is just perfect!’ ”

Jenna Bush Hager

Jenna Bush Hager Family CR: Jenna Bush Hager

Jenna Bush Hager Family CR: Jenna Bush Hager

Bush, 73, and his wifeLaura Bushwere “over the moon” to be expecting a grandson, Bush Hager shared with PEOPLE in an exclusive interview this past May.

“We had two girls. That’s all he’s ever known,” Bush Hager said. (In addition to having two granddaughters, her father is also a proud parent to two daughters of his own: Bush Hager and her twin sister,Barbara Pierce Bush.)

And although the then-mom-to-be said that her dadabsolutely adores “our girls,”she added that it would “be nice for him to have a little fishing buddy as well.”

Jenna Bush Hager Family CR: Jenna Bush Hager

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Bush Hager said during her call-in to theTodayshow on Monday that she’s feeling “good” and “happy” following Hal’s arrival via cesarean section. (Her newborn was born at 9:02 a.m. on Friday, weighing 7 lbs., 5 oz., and measuring 19.29 inches long.)

“I think when you’re held up in a hospital room with a newborn and you see the light in his eyes, you think, ‘This is what life’s for — to make the world better for our little babes,’ ” she said. “And even though thenews outside these walls isn’t so great,seeing the joy in their eyes, that’s what we’re here to do — make their lives safe.”

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Jenna Bush Hager

Jenna Bush Hager/Instagram

Jenna Bush Hager

Bush Hagertold PEOPLE following her son’s birththat he was named Henry “after many on his father’s side including his dear dad and his paternal grandfather,” while “Harold comes from my maternal grandfather, Pa, who I loved dearly and was unable to keep a son long enough to name.”

“His sisters are thrilled to have a little brother and already nicknamed his nickname, exclaiming, ‘Welcome to the world, Hal Pal!’ ” she added.

Bush Hagerrevealed her surprise pregnancy newson-air in April, nearly a week after co-hostHoda Kotbannounced she had adopted a second daughter,Hope Catherine.

Her pregnancy came more than three years afterHollywood MediumstarTyler Henrypredicted Bush Hager would one day have a son — to which she emotionally replied that she planned to name himafter her late grandfather Harold Welch.

source: people.com