WASHINGTON — It is a major headache for the national Republican Party and perhaps the biggest break Democrats have been handed in this difficult election year: a three-week runoff campaign in Mississippi between a party elder, Senator Thad Cochran, and the sometimes unpredictable Tea Party favorite, Chris McDaniel.

Mr. McDaniel, whose showing in the primary on Tuesday forced the runoff and shook the Republican establishment, carries the kind of baggage the party is eager to shed as it seeks to win over female and minority voters: He threatened to leave the country rather than pay reparations for slavery and described trying to pick up Mexican women by calling them “mamacitas.” He once dismissed a controversy over a wrestling video game in which a white woman holds down a black woman by shrugging, “Well, she wasn’t holding down a gay guy.”

Already on Wednesday, Democrats were quietly expressing glee and moving to elevate the McDaniel candidacy, hoping to make him this campaign cycle’s equivalent of Missouri’s Todd Akin, whose provocative comments on rape created problems for Republicans around the country in 2012.