TUCSON — Tucked in a teal booth at a local diner, beneath murals of an old copper-mining town, the southern Arizona political newcomer eagerly collected business cards from small-business owners over breakfast.

In a very purple district stretching east from Tucson to New Mexico, Lea Marquez Peterson, the Republican running for an open House seat here, needed every single one.

“I need all the support I can get, from every Republican entity out there,” she said in an interview afterward over coffee. “I’d love to see Hispanic voter turnout surprise everybody.”

In a year when a record number of women are running for Congress, Ms. Marquez Peterson, who leads the local Hispanic Chamber of Commerce, represents a rare combination for the Republican Party: female, Latina and conservative. Earlier this year, national Republicans were proclaiming Ms. Marquez Peterson as a new face for their future, along with two other House candidates: Young Kim, a former congressional aide who is running for a seat in California, and Maria Elvira Salazar, a Cuban-American journalist running in Miami. Both are in dead-heat races.