I want to add to those who mention the gilded age. There are some great opportunities to completely shake up the US political parties leading to a different 20th century.
What if Cleveland wins his home state of New York and the election of 1888? With a slightly better performance the Democrats could have held the House that year too, then they might be known as the first "billion dollar" Congress and lose big to the Republicans in the 1890 midterms. The McKinley tariff, Sherman anti-trust act, and Sherman Silver purchase acts would then be delayed as well as the admission of North Dakota, South Dakota, Montana, Washington, Idaho, and Wyoming as states. Then the Republicans could win in 1892 and be in charge when the Depression of the 1890's hits, leading to a Cleveland type Democrat winning the Presidency in 1896.
You could end up with a Democratic party that is controlled by the Bourbon Democrats, and have a longer lasting Populist party until they get absorbed by one of the big two. Perhaps if the Democrats become the party favored by big business the Republicans could turn more toward their progressive side to bring in some of the populists.