Chelsea Clinton has said the public's expectations of her mother were "very different" compared to her father.
The daughter of former US president Bill Clinton, and former presidential candidate Hillary Clinton, was speaking to the BBC after being named one of this year's BBC 100 Women.
"The expectations, at least publicly outside of our family, were very different for my mum versus my dad.
"No-one was asking him to change his name."
Ms Clinton said her father had always been supportive of her mother and her work, but added: "It would be disingenuous to say that her achievements were given the same clear-eyed assessment as my dad's were.
"I even remember in Arkansas in the 1980s, there was tremendous pressure for my mom to change her name from Hillary Rodham to Hillary Rodham Clinton, which she ultimately did because it seemed to her such a silly fight and she wanted to move that out to then focus on the work she felt called to."
Ms Clinton, who is vice-chair of the non-profit organisation the Clinton Foundation, said she had no immediate plans to run for office, but had not ruled it out in the future.
"It's always maybe, but it should always be maybe for everyone," she told BBC Radio 4's Today programme.
"While it's certainly no now, it's not a no-never, because I would hope that anyone who cares about the future of our country would feel that way whether they think about running for the local school board or president or anything in between."
She added that she hoped more people would think about running for office because it would "serve for us to have better choices and to able to make better choices at the ballot box".
Ms Clinton has written two children's picture books, She Persisted and She Persisted Around the World, which focus on women "who have used their voices and determination" to change and shape history.
One of the women in her book is Caroline Herschel, an astronomer who discovered eight comets in the 18th Century, but was less well known than her brother William, who discovered Uranus.
"We have long characterised how we have told history, taught history and come to understand and value historical figures," she said.
"I think it is hugely important, particularly now, that we shine a light on so many of the women who have helped to bring knowledge into the world and helped advance not only science but so many other areas of progress."
BBC 100 Women has profiled five women, including Caroline Herschel, whose achievements have been overshadowed by the work of male relatives.
What is 100 Women?
BBC 100 Women names 100 influential and inspirational women around the world every year and shares their stories.
It's been a momentous year for women's rights around the globe, so in 2018 BBC 100 Women will reflect the trailblazing women who are using passion, indignation and anger to spark real change in the world around them.