See
https://en.wikipedia.org/wiki/Walter_Q._Gresham for a brief biography.
It would have been fascinating (for me, anyway) to see him win the 1888 Republican nomination and go on to defeat Cleveland. Imagine a Republican president in 1889-93 who believed in tariff reduction and opposed imperialism and excessive spending. No billion dollar Congress, no McKinley Tariff, no US support for the overthrow of Liliuokalani. (No wonder that by 1892 Gresham had abandoned the Republican party and even flirted with Populism before deciding to back Cleveland...)
The only problem is that I have a hard time seeing him get nominated. True, he had advantages. Like Benjamin Harrison, he was from Indiana, a pivotal state in the Gilded Age (though as a judge on the Seventh Circuit Court of Appeals he lived in Chicago). He combined military, political, and legal experience. As a judge, he gained popularity with his decision in the "Wabash case" where he opposed Jay Gould's attempt to manipulate a railroad's reorganization. He had the support of the Illinois and Minnesota delegations at the convention, and especially of Joseph Medill's *Chicago Tribune* which represented western Republicans who were skeptical of the extreme protectionism of some Easterners. (Gresham was not a free trader. He had no problem with the protectionist plank in the 1884 platform. But he thought that the Republicans went way too far in 1888 when their platform called for addressing the surplus by doing away with *all* federal internal taxes before making the slightest dent in the protective system...) He was the favorite candidate of the Mugwumps, many of whom were somewhat disillusioned by Cleveland's record on civil service (though Cleveland's 1887 call for lower tariffs did re-kindle some of their support). On the first ballot at the convention, he finished in second place, behind John Sherman.
But the obstacles seem just too great:
(1) Harrison won control of the Indiana delegation in his struggle with his fellow Hoosier Gresham.
(2) The Blaine supporters much preferred Harrison to Gresham. (Indeed, many of them at first backed Harrison simply to block the front-runner John Sherman and to create a deadlock which they assumed would be resolved by a stampede to Blaine. When Blaine definitely ruled out accepting the nomination, they found Harrison a satisfactory substitute.)
(3) The very fact of Gresham's support by the Mugwumps and the *Chicago Tribune* made some Easterners shudder. (True, Gresham also had the support of the *Tribune*'s rival the *Chicago Inter-Ocean* which claimed that Gresham was "as good a protectionist as Blaine." Charles W. Calhoun, *Gilded Age Cato: The Life of Walter Q. Gresham*, p.96
https://books.google.com/books?id=otUeBgAAQBAJ&pg=PA96 But, as Calhoun noted, this was not very reassuring, because Gresham refused to dicuss the matter publicly; his associates merely referred questioners to a mildly protectionist speech he had given in 1884--and if anything he had moved away from protectionism in the intervening years.)
(4) In spite of all this, Gresham might have had a chance if Thomas Platt, the Stalwart boss of New York, would support him. (Platt's first choice was nominally Chauncey Depew, but nobody took Depew's candidacy seriously. The question was who Platt would switch to. He definitely did not want John Sherman, who as part of the Hayes administration had participated in the purge of the Stalwarts from the New York Customs House.) Platt actually did make overtures to Gresham before and during the convention, saying that Gresham was the most electable candidate. (There were concerns that Harrison's vote against the Chinese Exclusion Act of 1882 might hurt him with the labor vote.) Gresham's widow Martha Gresham later claimed in her biography of her husband that Gresham could have had the nomination if he had been willing to promise Platt the Secretaryship of the Treasury. At one point Platt apparently also suggested a Gresham-Depew ticket. Most likely, though (as Mrs. Gresham recognizes) the overtures were really a feint to pressure Harrison into promising the moon (not to mention the Treasury...) to Platt.
As it turned out, Gresham did not get a single vote from either the New York or Pennsylvania delegations in the convention, despite the attempts by Gresham's backer John W. Foster to reassure Platt and Pennsylvania's boss Matthew Quay that they would be treated fairly in matters of patronage.
Probably to get Gresham nominated we have to have his Indiana rival Harrison somehow eliminated from the race. (One other possibility: Make Blaine a candidate. This will *probably* lead to Blaine's nomination but instead it just might lead to Gresham as the successful candidate of an "anyone but Blaine" movement.)
(Of course there is another way of getting Gresham to be president: Have Cleveland die in surgery
http://www.npr.org/2011/07/06/137621988/a-yacht-a-mustache-how-a-president-hid-his-tumor and his successor, the first Adlai Stevenson, also die somehow. Gresham as Secretary of State is next in line under the 1886 Presidential Succession Act. But given that he died in 1895 in OTL, his presidency under those circumstances would presumably be brief...)