"Running" also meant something very different in 1968 than it does today. There were only a few primaries, and the party bosses still controlled a majority of delegates. At that time, there were three main strategies that a candidate could use to win the nomination:
1-Strictly convention route (Humphrey '68, Johnson '60): This strategy is employed by candidates who are strongest among the party leadership and bosses. They rely upon the support of these "king-makers" for the nomination. This route is chosen by candidates who either a) have a strong base of support among delegates or b) would struggle in the primaries. For this strategy to be sucessful, the primaries must be devisive and inconclusive. But if a candidate (such as Kennedy in 1960) is able to dominate the primaries, this convention route is likely to fail.
2-Strictly primary route (Kefauver '52/'56, McCarthy '68): In contrast stark contrast to the convention strategy, the primary route is one entirely focused on winning the primary states. That's not to say that a candidate wouldn't try to win over bosses or unpledged delegates. What it means is that the candidate has practically no appeal to them or chance of winning on the convention floor. These candidates need to sweep the primaries to have any chance, and even then they usually come up short. The hope is that their momentum will convince the leadership of their superior electability. These candidates tend to be on the fringes of their party or a personally polarizing figure.
3-Balanced route (Stevenson '56, Kennedy '60, Humphrey '72): This approach is a combination of the first two. It is also the most difficult to organize. Candidates who employ this strategy compete in the primaries while also working to attract support from the bosses. To do requires money and planning. Kennedy is the best example of this strategy. He spent 1958 and 1959 flying around the country with Ted Sorensen feeling out support and getting to know the various state and regional leaders. He then ran hard in a few select primaries where he could prove his electability. This strategy can only work with a candidate who appeals to both voters and leaders. Otherwise, as was the case with Humphrey in '72, it falls flat.
So acknowledging that these were the three principle ways to win the party nomination, here's how I think the various Democratic candidates would run.
Vice-President Lyndon B. Johnson (convention-only): I don't see Johnson campaigning for the nomination. He'll only take it if it's given to him; partially because of his ego, partially because of his health. This would require President Kennedy not attempting to derail Johnson's candidacy. If he actively works to persaude bosses to oppose his VP, than Johnson's probably not going to win the nomination. His base of support will likely be the Southern delegations who still see him as more of a moderate than a liberal.
Senator Hubert H. Humphrey (balanced): Humphrey would need to run in the primaries to gain enough traction to win the nomination. His appeal with organized labor, African-Americans, and liberals will be quite high. But the more conservative urban bosses and Southern delegates will need to be persauded. That involves winning a few big primaries, and also persuading folks like Daley to back his candidacy. This could very well work out for him if Kennedy works to stop a Johnson nomination.
Secretary Robert F. Kennedy (convention-only): RFK is not going to be the Democratic Party's nominee for President in 1968 unless a few things happen. First off, there cannot be a clear primary winner. Second, his brother is going to have to work against Johnson and for Bobby. Third, their needs to be a deeply divided convention which will rally to the family name. But unless also those things happen, and potentially even if they do, than there's not going to be a Kennedy in the White House come 1969.
Senator Eugene McCarthy (primaries-only): I could see McCarthy running in 1968, especially if the Vietnam War still occurs in a somewhat simmilar way to OTL (a view I now hold). He'll be the anti-war candidate, espousing a view simmilar to that in OTL. The biggest difference is that he won't be going against a sitting President or a Kennedy. This gives McCarthy a great oppurtunity to win a few more primaries against Humphrey or whoever else might run. But he'll still have to overcome a party establishment that doesn't want to nominate an outsider and maverick. He'll have to sweep the primary contests if he wants any chance to win, and even then it's unlikely.
Senator George McGovern (balanced): If McCarthy doesn't run (unlikely), than McGovern could very well become the anti-war candidate in the race. He didn't run in '68 largely because he didn't want to challenge a sitting President or go against his friend, Robert Kennedy. Unlike McCarthy, McGovern had a degree of support within the Democratic establishment. He wasn't nearly as enflamatory as McCarthy in the Senate and had cultivated a positive relationship with other liberals in the body. He'll have to run and win a few contests, but can also expect some support on the convention floor.
Senator Abraham Ribicoff (convention-only): Ribicoff, despite being a fairly unknown figure in the 1960s, was actually a very important force in 1960s American politics. He was a Governor of Connecticut from 1955-1961, Secretary of HEW from '61-'63, and then Senator from '63 onward. If there's a brockered convention, I can see Ribicoff becoming a favorite of Northeastern Democrats. His religion would probably prevent him from winning the nomination, but he could make a run at it.
Governor John Connally (convention-only): If Johnson refuses to accept the nomination, why not have his chief lieutenant run? Connally was a conservative on social and buisness issues, but was by no means a full-blown Dixiecrat. Unlike George Smathers, another moderate Southerner, he wasn't closely tied to Kennedy. That's probably a positive for Southern delegates who would want a break with the past President. Connally probably ends up being the sacrificial lamb for the South, but could very well end up on the bottom of the ticket.
Secretary Robert McNamara (convention-only): McNamara was allegedly Kennedy's top choice to run for President in 1968. This might change after five more years in the White House and an unpopular war in Vietnam. But McNamara was still an immensly qualified candidate. He would make an excellent compromise candidate. The only major caveat is Vietnam. If it's going poorly, than McNamara's going to have a hard time wooing the anti-war vote and party bosses worried that a nominee so closely affiliated with the war would be headed for defeat.
My personal prediction would be that Humphrey, McCarthy, and various favorite son candidates fight it out in the primaries. Humphrey probably wins most of them, but McCarthy has a good chance of winning a state like Oregon. If that's the case and there's no strong candidate coming out of the primaries, than the contest goes to the convention floor. At that point, you're going to see many bosses look to Johnson. Kennedy may attempt to deny him the nomination, but I doubt he'll put that much effort into it. His brother on the other hand very well might. But with the chance of a protracted convention, I see enough delegates rallying behind the Vice-President for him to accept the challenge and go for it. At that point, he probably wins the nomination. His running-mate will probably be a Northern liberal, such as Humphrey.