The obvious answer is Senator Alben Barkley of Kentucky, but any campaign he would mount may very well run into the same issues he experienced when he tried to run in '52; he was 71 after all, which back then was still a height that caused some concern among both the voters and the delegates. Certainly there would be comparisons with Roosevelt and the need to get his running-mate right for those who decided to back him.
Senator Claude Pepper could make a go at it, though I have doubts he would get all that far given he also would be facing the same concerns he faced in OTL, namely his dovish positions in regards with how to deal with the Soviet Union.
Associate Justice William Douglas as others have put forward seems a strong candidate and he might very well have received the support of Truman for the nomination, but his nomination would be very divisive. The Southern delegations would be strongly opposed to his nomination by virtue of his more Liberal temperament, and it is not clear whether he could unify the the more Progressive Democrats behind him even with establishment support.
Richard Russell would serve as the nominee of the Southern delegations much as he did in OTL, but he has no chance of being nominated.
Unfortunately not many other candidate comes to mind, at least those that would have elicited more than a handful of delegates.