The 53-year-old actress discussed her exchange with the British royal during Tuesday's episode of Watch What Happens Live.
"I wasn't that drunk, but enough drunk not to drink anymore," Bonham Carter said. "He wasn't totally sober himself. Not in a bad way; it was just really jolly."
Bonham Carter has two children, 16-year-old son Billy and 11-year-old daughter Nell, with her ex-partner, Tim Burton.
"The thing is, I'd spent all my friends as godparents on my first child," the star said. "Billy had 11 godparents, and I thought I could make up for Nell ... I thought, 'Well, if I did for Prince William, that would cover it.'"
"He just said, 'You don't want me to be godfather to your child,'" she recalled. "It didn't work out."
Bonham Carter initially shared the story at a charity dinner in 2014.
"I got very drunk at one point, and I was determined to ask him to be a godfather to my daughter,'" she said of William. "He quite wisely said no."
"I don't pay attention to it, and I wish she wouldn't have to," the actress said. "I feel like there's just so much. It's just all too much."
"As soon as you become well known and in the public domain, you have to kind of accept that all that people think about you, write about you and whatever -- that's about them, that's not really about her," she added. "Why ... put your self-esteem in the hands of strangers."
Carter plays William and Harry's great-aunt Princess Margaret in Season 3 of the Netflix series The Crown, which premiered this month. She will reprise the role in Season 4.