Really, leaving aside butterfly effects, it is hard to see how the 1880 election made much difference. The tariff? The Democratic platform was ambiguous, and Hancock himself strongly denied that he would end protection for industry. (Unfortunately, he said the tariff was a "local question." He later explained privately that what he meant was that it affected different localities differently, but his statement was still ridiculed by the Republicans.
http://books.google.com/books?id=ubSem4UEn9AC&pg=PA301) Civil service reform? All the candidates said they were for it, and it would come eventually, but probably Congress would do nothing about it in the near future. (The only reason it did so in OTL was because of Garfield's assassination. So, yes, in that sense Garfield's election made a bit of a difference, but only accidentally.) The money question? Hancock was not as prominently associated with hard money as Tilden and Bayard were, but there is little likelihood he would have aligned himself with western inflationists.
Hancock's election would not be good news for African Americans, of course. His biographer David M. Jordan writes "And to the black American. Hancock apparently gave little thought at all, once the institution of slavery had been crushed and the Union restored."
https://books.google.com/books?id=ubSem4UEn9AC&pg=PA318 Yet, Jordan also notes that Hancock was almost alone in the Army in supporting the rights of a black cadet in a scandal at West Point. And in any event, since 1876, the Republicans had pretty much abandoned African Americans, so it is hard to see Hancock's victory making much difference on racial maters, except perhaps fewer patronage jobs for African Americans--and yet they might still get some, as they did under Cleveland.
Leonard Dinnerstein in “The Election of 1880,” in A. M. Schlesinger, Jr., ed., History of American Presidential Elections , 4 volumes. New York : Chelsea House and McGraw-Hill, 1971) describes the 1880 election as "one of the most insignificant in United States history." It certainly did not seem that way to voters at the time--turnout has been estimated at 78 percent. But in terms of likely actual effects on public policy, I would say Dinnerstein was right.