Has anyone from a President's own party ever run against a sitting President in his second term election?
Certainly.
1968: Sen. Eugene McCarthy and Sen. Robert F. Kennedy opposed President Lyndon Johnson's re-nomination. Shortly after the New Hampshire Primary, Johnson quit the race. Several months later, Robert Kennedy was assassinated, and the party turned to Vice President Humphrey who was defeated by Richard Nixon in the general election.
1976: Former Governor Ronald Reagan opposed Gerald Ford's efforts to win nomination for a full term. (Recall he had taken office as a result of Nixon's resignation.) Although it was a close contest, Ford prevailed and went on to lose in the general election to former Governor Jimmy Carter.
1980: Senator Edward Kennedy opposed Jimmy Carter's re-nomination. Carter won re-nomination but was defeated in the general election by former Governor Ronald Reagan.
1992: Former Nixon adviser and television commentator Patrick Buchanan opposed President George HW Bush's re-nomination. Bush won re-nomination but was defeated in the general election by former Governor Bill Clinton.
Note that each time an incumbent President faced a serious primary challenge, it left his party so divided that he (or in the case of 1968, his Vice President) lost in the general election.
Technically this would be LBJ’s third term, but a “Dump Johnson” campaign had sprung up in 1967, seeking to oppose President Johnson as the 1968 Democrat Nominee for President. The Dump Johnson group first asked RFK. They had a series of meetings in September and October 1967 before Kennedy turned the group down The Dump Johnson movement would ask five more people to be their standard bearer, including Senator Frank Church of Idaho and Senator George McGovern, before finding Eugene McCarthy of Minnesota who said yes.
McCarthy was underfunded, even thinking of by passing the New Hampshire primary for the Massechuett primary the week after. A reporter asked Gene if he was committing political suicide taking on LBJ’s control of the party apparatus. Wait until the coffins start coming home to small towns in America, you see the people rise up against this war, said McCarthy.
McCarthy didn’t set foot in New Hampshire until January 25, 1968. He was polling 12%. The McCarthy campaign was being written down as a soon to be forgotten footnote in campaign history when the January 31 Tet offensive and uprising exploded in South Vietnam. All the US government and military leaders claims that the enemy was almost defeated shocked the American people. Young people flocked to New Hampshire to join Eugene McCarthy’s campaign, including a young Wellesley student named Hillary Rodham.
Many hippie styled youths shaved their beards, going “Clean for Gene,” in order to entice voters to back their candidate.