Katherine Schwarzenegger shares Maria Shriver's reaction to her grandchild's name

Katherine Schwarzenegger carried on a proud family tradition that left her mother "very pleasantly surprised" when she found out the name of her first grandchild.

Schwarzenegger and husband Chris Pratt named their first child Lyla Maria Schwarzenegger Pratt, with her middle name paying homage to Schwarzenegger's mother, NBC special anchor Maria Shriver.

"Well, we didn't know if it was gonna be a boy or a girl, so I always knew that I wanted to incorporate my mom's name if it was a little girl because I'm Katherine Eunice, so I'm named after my grandma (Eunice Kennedy Shriver)," Schwarzenegger told co-anchors Savannah Guthrie and Hoda Kotb on TODAY Tuesday. "So I wanted to definitely continue that tradition on, but I think she was very pleasantly surprised with the name."

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Schwarzenegger, 31, is adjusting to life as a new mom after her daughter's birth last August.

"It's the best thing in the entire world," she said. "I love every minute of motherhood so far, and I just feel really blessed and really lucky 'cause she's such a happy and easy baby. So my mom continues to remind me of that."

Related: Schwarzenegger is expecting her first child with husband Chris Pratt.

Shriver is not only soaking up every moment of being a grandmother, she's also been a sounding board for her daughter.

"Every morning I wake up with a newfound respect for my mom," Schwarzenegger said. "I've always respected her tremendously, but I even texted her this morning because I was woken up this morning at 2 and I had my alarm going off at 3 to get ready for this, and I was just like, 'I can't believe you did this with four children.'

"And she does everything so beautifully and gracefully so I always have gained more respect for her, but I call her every single day with something, so it's just such a fun experience and a great time so far."

Shriver shared her anticipation about becoming a grandmother on TODAY in July.

"I'm excited, I'm nervous," she said. "I don't know what it's going to be like, but I think I'm going to be a great grandmother because ... I'm gonna like spray the sprinkler, I'm gonna do the three-legged races. I'm gonna hopefully have a partner in crime again."

Related: Shriver and Pratt had a heartwarming conversation that showed the love runs deep in their family.

She has lived up to that billing, Schwarzengger told Hoda on TODAY with Hoda & Jenna Tuesday. Schwarzenegger's father, "Terminator" star and former California Governor Arnold Schwarzenegger, has also become a doting grandparent.

"Oh, my gosh, she's so fun," Schwarzenegger said about Shriver. "She's the most fun grandmother ever, and she has a really fun time doing it. It's been such a joy for me to be able to see both of my parents step into the role of those grandparents and to see my two brothers become uncles and my sister become an aunt and my husband become a girl dad."

Schwarzenegger also shared what she hopes to pass on to her daughter after growing up herself as the child of two famous parents.

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"I think obviously I didn't grow up with social media being a thing at all so it's a little bit different in that way just because I feel like we share so much in today's world, but I think one of the greatest gifts my parents ever gave me and my siblings is the gift of privacy and having a really normal upbringing, or as normal an upbringing as possible," she said.

"We had a really normal and magical childhood and we were allowed to kind of be our own people and have our own identity and kind of choose to step into whatever form of being public we wanted to when we felt comfortable. And I think that was such an incredible gift to have given us kids, and so it was something that was really important to my husband and I to be able to give to our kids and be able to have that privacy and not necessarily show as much of them on social media."

On Tuesday, Schwarzenegger was also celebrating the paperback release of her fourth book, "The Gift of Forgiveness," which features in-depth interviews and stories about forgiveness as well as Schwarzenegger's own journey with it.

"When I was struggling with forgiveness in my own life, I started talking to different people about their struggles with forgiveness and their experiences with it and realized how beneficial it was for me to hear other people's experiences with forgiveness and be able to apply that to my own life," she said.