OK.
All you'd need to do is turn the Democratic Party into a fringe party like the Socialist Worker's Party and then it can be a "black person's Party" of some sort.
So, what you could have is the Populist Party and the Republican Party become the dominant parties in American Politics, because William Jennings Bryan defects to the Populists and the Democrats of the late 1890s get polarized into Republicans and Populists.
The Democrat Party fades into a whimper, but the name is revived in the 1960s when American Blacks, seeking political mobilization, bring the Democratic name into a modern political environment, much like a Black Person's party.
Al Sharpton, a leader of the Black Community, easily gets the nod, maybe with MLK III as his running mate. They have no chance to win, but if they can influence the election or shift things in their own favor by any margin, they have succeeded.