Fox News did not respond to requests for comment.

Her early climate work included taking a position in 2008 with the newly formed Clinton Climate Initiative, part of the Clinton Foundation established by the former president. As for her family connections, “I was certainly aware that the other Murdochs are conservative,” the founder of the climate initiative, Ira Magaziner, said, but “it didn’t matter.” At the time, he noted, many Republicans, including Mitt Romney, had spoken about the need to deal with climate change.

Over the years, she has worked with organizations whose approaches to environmental action align with her own. She joined the board of the Environmental Defense Fund, a group that often collaborates with industry on climate issues. In 2014 she and her husband created the Quadrivium Foundation (the word means “crossroads”) to fund their programs. One of those is SciLine, an independent nonprofit service that connects reporters with scientists and provides fact sheets on topics in the news like a primer in August on hurricanes and climate change.

She decided, however, that spreading scientific knowledge might not be enough. People already understand that the planet is warming and that humans are the cause. The deeper problem, she said, is that the government of the United States isn’t doing anything about it.

She took a deep dive into possible solutions to partisan deadlock and reviewed the players in the diffuse field known as democracy reform: Small groups that push for changes in the electoral system.

Some of the avenues her groups are pursuing include ranked-choice voting, in which voters rank candidates in order of their preference. Proponents of this method argue that it reduces the tendency of primaries to reward candidates who work mainly to energize their base, and favors candidates who have the broadest appeal. She is also interested in initiatives to restrict gerrymandering and increase access to voting through proposals like automatic registration, as well as open primaries, in which voters do not need to declare their party affiliation.

Charles Wheelan, founder of Unite America, said that because of Ms. Murdoch’s surname, “it’s fair to say that in some quarters our relationship raises a few eyebrows.” But he also calls her “an important ambassador” to the wealthy and powerful, someone who can tell them, as a peer: “Look, if you really want to make the world better for your grandchildren, fix politics.”