For years, fans of music icon Beyoncé Knowles-Carter and Shawn "JAY-Z" Carter have wondered about the true story behind their daughter's distinctive name, Blue Ivy. 

The speculation and myths finally met reality when Carter told "CBS Mornings" exclusively that the name wasn't a color preference or a carefully orchestrated branding strategy, as rumors had suggested, but rather a playful, affectionate term that blossomed from the sonogram's screen.

"It was supposed to be Brooklyn," Carter told King, nodding to the deep personal connection the two feel with the New York City borough where he grew up.

But during their routine prenatal checks, a different inspiration struck the couple.

"We was calling her Blueberry. Like, 'Look at the little blueberry.' You know, it was like a nickname," said Carter. "It just was natural." 

Eventually, the affectionate nickname that accompanied every sonogram and doctor's visit began to feel more like a name than a nickname, Carter said. "We just took the 'berry' off of it and called her Blue."

Eleven years later, Blue Ivy is carving her own path in the music world, taking center stage next to her mother and performing as a dancer during Beyoncé's "Renaissance" world tour.  

Carter, who chose to step back from his typical on-stage role, watched from the sidelines. 

"For me, this was her best tour," he said. "This one to me felt like the most complete." 

While performing in front of thousands of people is nothing new for Beyoncé and JAY-Z, it was for Blue Ivy. He said he watched with "goosebumps" as, despite Blue Ivy being nervous, she was determined to face the crowd.

"Blue's been born into this world that she didn't ask. She's been born into a life she didn't ask for. So since she was born she's been in, like, scrutiny and public eye, and everyone having an opinion," said Carter.

"She wanted to do it. She wanted to do it the first night, and we was like, 'OK, if this is something you wanna do, you can't just go out there. You gotta go work with the dancers, and go work. And she worked every day and watched her work hard," he said.

The song "My Power," which was featured during her performance, felt like a defining moment for Blue Ivy, said Carter. "You can't write a better script."

King and Carter spoke about his family during a wide-ranging interview and tour of the "Book of HOV" exhibit at Brooklyn Public Library. He said he and his grandmother shared an emotional experience while sitting in one of the exhibit's screening spaces on opening day.

"My grandmother is the reason why I'm, like, reserved and holding my feelings in, it comes from her to my mother, to me. And she started crying." After that, the rapper said he started crying, too.

"And I've maybe seen her cry three times in my life," he said. "She just, you know keeps everything close to the vest. That's who she is. I got a lot of my ways from her. And my mother got a lot of ways from her as well. I see it now."