The best recipe, IMO, is to keep Cleveland out of office long enough to get a bit more politically skilled and to attract a more defined political following / program. The fact that Cleveland was actually a really upright guy might also contrast well with some of the nastiness of the times:
First, we must prevent Cleveland from gaining the Democratic nomination in 1884. Say Allen Thurman takes it, perhaps. We can posit Cleveland being absent from some crucial meeting or some such. Thurman then loses to James G. Blaine. Cleveland then stays of Governor of New York. He is forthright reformer, per his OTL record. (Cleveland was Governor of NY only for about 2 yrs before becoming President).
Blaine's presidency goes horribly awry: corruption looms and seems to blossom. The debate over silver becomes more heated. Labor unrest is a bit more marked. However, Blaine manages to win in 1888. He pushes an aggressive foreign policy to win. Things continue to get worse.
Convinced of the need to restore right conduct to the White House, Cleveland runs in 1892, winning as the Republicans fragment in the far West. Cleveland is remembered as a great reformer, introducing the principle of non-partisan appointments in the federal bureaucracy. He busts trusts (though less than TR did OTL) but also prevents labor disputes from becoming violent. He wins re-election in 1896, as his "Reform Democrats" have a wide following. Constitutional Amendments to grant the President a line-item veto in appropriations and to ensrine civil service procedures pass in 1898. Hawai'i becomes a proctectorate of the US (Blaine had pushed the planters into an early less successful revolt, but Cleveland's support for Queen Liliuokalani wins the US prestige). The Cleveland Corrolary of the Monroe Doctrine finds expression in Cleveland's pressing for arbitration and presiding over mediation between Cuban rebels and Spain. The US assumes part of Cuba's debt in exchange for basing rights at Guantanamo and guarantees Cuban independence.
These payments are eased by the financial health of the US, after successful negotiations in Congress resulted in the Wilson Tarrif of 1896 -- at 25% the lowest tarrif in recent memory. Bilateral agreements with Canada soon follow, fostering Anglo-Candian coopeartion.
Cleveland reluctantly runs for a third term in 1900, as the Cuban Mediation is still on-going. He wins, making history.
He is remembered for his consumate rectitude in office, his honesty, and his commitment to non-intervention as the best way to uphold the interests of the US and its values.
His second Vice-President, William Jennings Bryan, wins election to replace him in 1904, as the Republicans had finally split under Teddy Roosevelt's Progressives. Bryan proves unable to deliver on Cleveland's legacy, his one accomplishment--financing of the Nicargaua Canal--forced on him by Congress and opposed to his own principals (much like Jefferson and the Louisiana Purchase).
Teddy Roosevelt's victory in 1908 serves in many ways to confirm Cleveland's legacy, a fact the first Progressive President noted in his memoirs, having served as an assemblyman while Cleveland was Governor: Roosevelt too helps to pass Constitutional amendments that reformed American government for the better (the 17th, introducing the direct election of Senators, and the 18th, introducing the income tax and outlining the obligations of the federal government vis-a-vis the economy and thus tacitly enshrining the Square Deal into Constitutional Theory as well as the 19th, introducing Women's Suffrage). Roosevelt confirms precedent of serving three terms, taken from granted in the 20th Century. Roosevelt's connection to Cleveland's diplomatic legacy is more difficult to perceive, given Roosevelt's more forcefull character. Nonetheless, the Ultimatum of 1915 (a promise to attack any aggressor) holds out a firm hope of peace in the descent that leads to the Great War. While Roosevelt's dictum did not dissuade Wilhelmine Germany, his firm prosecution of the war and his able management of the Portsmouth Conference that ended it serve as testament to Cleveland's belief in the need for a "balance of peace."
Probably works better if TR isn't also successful (three-way politics end up making the 1910s-1920s much more contentious, say), but I couldn't help myself. I'd guess Progressives and Democrats will debate between the two much as we do between Wilson and Roosevelt or between FDR and TR, perhaps. Probably most people will hold out that someone else is the greatest, but Cleveland will have the virtue of being the only one not to have been a wartime President (much as TR does today).
) make Grover Cleveland a serious rival with the likes of Lincoln and Washington for the Greatest US President?