Morris K. Udall: Made a serious run for the Democratic nomination in 1976, losing to Jimmy Carter. By that time, though, Udall had distanced himself from Mormons:

During the Michigan primary, the Carter campaign had Coleman Young, the mayor of Detroit, accuse Udall of racism for belonging to the Mormon church, which at the time, did not allow blacks to serve in the church's priesthood (since changed in 1978 by revelation to the Mormon prophet, Spencer Kimball). Young's attack was at least somewhat unfair, since Udall had been a longtime critic of that church policy, and had ceased being an active member because of it. Carter's subsequent sweeping of the black vote in the Michigan primary was key to his crucial victory in Michigan.

George W. Romney: Many know about Mitt Romney’s father, three time governor of Michagan. He was a serious candidate for the 1968 Republican nomination until he claimed being “brainwashed” about the Vietnam war. An interesting note from Wikipedeia is that George Romney was born in a Mormon colony in Mexico:

Romney was born in Mexico, which raised the issue if he was eligible to be President, which is constitutionally limited to a "natural born citizen" of the United States. Both his parents were American citizens, and he returned to the U.S. before he turned 21. That was sufficient for him to be a U.S. citizen, but not necessarily to pass the "natural born" test (which only applies to presidents). The issue was never tested in court and contrasts with the case of John McCain, who was born to American parents in the Panama Canal Zone at a time it was part of the U.S. and his father (a naval officer) was assigned to duty there. The Romneys had purposely left American legal jurisdiction.

Finally, the first Mormon, Joseph Smith, declared himself a candidate for president in 1844 (he was killed later that year). Smith was probably not a serious candidate, his purpose may have been to highlight the petitions for redress due to Missouri’s expulsion of Mormons six years earlier.