I reckon you'd need a two-fold POD. First off, have the Democratic Convention contested seriously by O'Conor or perhaps someone else (anti-Catholic sentiment was still too high IMO to justify a Catholic Presidential Candidate) or have a rift develop between the Liberal Republicans and the Democrats (more likely IMO). Thus the election of 1868 sees the Democrat Party split with a much larger Straight Out Democrat following under O'Conor or perhaps Jermiah Black...
Grant still wins the election by a landslide despite many Democrats jumping ship and joining the Liberal Republicans.
The next POD you'd need is to have Horace Greeley survive. His death pretty much killed the Liberal Republican Party. He lives longer, so does the party, and by extension the rift within the Republican Party. He died fairly young so giving him another 5-6 years wouldn't be too much of a stretch, any more...
Then to finish things off you'd need the remaining Democrats to be completely stupid and refuse to nominate Greeley and continue on their states' rights anti-reconstruction bent. Then have Grant successfully run for a 3rd term and give Greeley some kind of celebrity VP (Winifield Scott Hancock perhaps?). Grant ends up being defeated by Greeley who enters the White House with a surge of support. The election of 1876 would mark the last election in which the Democratic Party would win any seats (the LR's taking over in the south) though it would continue to nominate candidates for the next two major elections.
Given his poor health, Greeley probably dies in his first year as President leaving his celebrity VP (again Hancock perhaps the best choice) as President. Championing Liberal Republcian values Hancock goes on to win re-election in 1880 serving nearly 2 terms before stepping down.
Longer term you'll probably see the Liberal Republicans get hijacked by the Populists like their OTL analog the Democrats...However I'd wager that the Supreme Court's going to be considerably more liberal than in OTL perhaps leading to a different ruling on the Homer vs. Plessy case. No Segregation perhaps?