EMMETT, Idaho — This state is indisputably Trump country, what with the president’s 32-point victory margin in 2016 and the Republican state leadership and congressional delegation. Still, in storefront windows like at the Possumtrot Traders pawnshop in this deeply conservative town of 6,800, you see it: a large Paulette Jordan sign displayed next to a Trump sign.

Ms. Jordan is the Democratic nominee for governor in Idaho. But Tuesday’s election, which she is not likely to win, is not the only reason she is interesting. She also represents a fresh attempt by Democrats in the frontier West to find a brand of politician who can help keep the party credible and energized, instead of an endangered species. In other governor’s races in the region, Democrats are running a Latino educator in Arizona, a gay congressman in Colorado, a white businessman in Nevada and prominent female leaders in New Mexico and Wyoming.

A 38-year-old former state legislator, Ms. Jordan holds some traditional Democratic positions like helping low-income Idahoans, supporting abortion rights and decriminalizing marijuana. But she also wants to give local officials more autonomy on tax policy and land use, and supports term limits. She owns guns, and while she supports universal background checks, restrictions for domestic abusers and a ban on bump stocks, she does not advocate stricter measures like a ban on AR-15s — the weapon that is popular with many gun owners but also used in many mass shootings, including last Saturday’s attack at a synagogue in Pittsburgh.