Senator William Borah (R-Idaho), the famous progressive and arch-enemy of the League of Nations, was often mentioned as a possible presidential candidate, but did not actually enter the GOP primaries until 1936, when he lost to Alf Landon. Even had he been nominated in 1936, his chances against FDR would have been negligible (yes, even if Huey Long had lived).
However, there was one time Borah actually had a chance to become president. In 1924, Coolidge wanted Borah as his running mate, in order to counteract La Follette's appeal in the West. Borah declined, preferring to remain in the Senate where he was next in line for the Chairmanship of the Foreign Relations Committee after Henry Cabot Lodge (who, as it turned out, was to die later in 1924). The US Senate's online biography of the man actually chosen for vice-president, Charles Dawes, gives the following account: "Nor did the popular Idaho Senator William E. Borah want to be the number two man. A story at the time recorded that President Coolidge had offered Borah a place on the ticket. 'For which position?' Borah had supposedly replied."
http://www.senate.gov/artandhistory/history/common/generic/VP_Charles_Dawes.htm
Now suppose Borah had accepted, and the Coolidge-Borah ticket would of course be successful. Suppose Coolidge dies while president. A few ways this could happen:
(1) Although Coolidge didn't die until January 1933, he may have had a heart condition for some time before that; the tiredness and breathing problems of the last period of his life, which he credited to the catch-all illness of the time, asthma, may have been cardiac asthma--heart disease with failing circulation. "The biographer [Claude M.] Fuess said in 1962 in a tape-recorded oral history that Coolidge had had a heart attack during his presidency and knew that his days were numbered." Robert H. Ferrell, *The Presidency of Calvin Coolidge*, p. 204. Perhaps this was one reason he "did not choose to run in 1928."
One may of course ask why there would be a *fatal* heart attack in this TL when there wasn't in OTL, but Borah had a reputation as a contrarian--according to
http://www.kevincmurphy.com/williamborah.htm one of his nicknames was "The Great Opposer"--and one can imagine Coolidge feeling under particular strain after a heated argument with his vice-president over, say, the situation in Nicaragua (see below).
(2) If there's no fatal heart attack, we can have Coolidge assassinated, maybe by a "crazed anarchist" upset over the Sacco-Vanzetti case. (Again, why not in OTL? Well, maybe this would-be assassin in OTL is discouraged by the thought that Dawes might be at least as evil a capitalist lackey as Coolidge but in this ATL thinks that Borah, with his somewhat "progressive" reputation, will at least be a lesser evil than Coolidge. Of course some anarchists would argue that "the worse the better" but this particular assassin does not see things that way.)
(3) Of course there's always the possibility of a really freakish death--e.g., Coolidge is playing tennis, rubs a blister on his toe that becomes infected, a rare form of blood poisoning sets in, and he dies. (Actually this did happen in OTL in 1924--to Calvin Coolidge *Junior*, the president's 16 year old son.)
Anyway, Borah becomes president. On domestic issues, there is presumably stricter antitrust enforcement than under Coolidge; Borah's progressivism was of the old-fashioned anti-monopoly kind. It is also possible that as a westerner with that region's distrust of Wall Street, he might show greater concern about the speculative stock market boom than Coolidge did. Ironically, given Borah's reputation as a farm state progressive, there is one issue on which he was to the "right" of Dawes: Dawes supported the McNary-Haugen bill, Borah opposed it. (McNary-Haugen was a scheme to help farmers through a government corporation which would buy up their surpluses and dump them abroad or keep them off the market until prices rose. It was very popular among farmers, though economists generally agreed with Coolidge that it would not work, and that the guaranteed higher prices would just encourage larger surpluses.) Borah opposed it, as he would oppose the New Deal's AAA in the 1930's. See
http://partners.nytimes.com/library/politics/camp/280615convention-gop-ra.html for an account of Borah's speech in opposition to McNary-Haugen at the 1928 Republican national convention:
"But he was firm for what he contended -- that the McNary-Haugen principle was unconstitutional and that the farmers would fare better under the form of less radical measures of agricultural relief proposed in the plank sponsored by the majority of the Resolutions Committee than by any further effort to enact the equilization fee experiment which the McNary-Haugen measure contained....What attracted most attention in Borah's speech was the high praise he bestowed on President Coolidge. This independent from an agricultural mountain State, the man whose Republicanism has at times hung by a thin thread, who has angered the party regulars by his insurgency, became a regular himself in that brief period he appeared before the delegates of the party."
One group that definitely would not benefit from a Borah presidency would be African Americans--at least so far as their civil rights were concerned. Borah helped to defeat the Dyer anti-lyncing bill, and justified southern disfranchisement of African Americans.
https://thebluereview.org/william-borah-lynching-history/ One should note that many other western Progressives held similar views--they wanted to leave the South alone to do what it wanted with its African Americans so that the South would back the western states against the "yellow peril."
In foreign policy, Borah is often regarded as a mere negativist. (Coolidge is supposed to have once expressed amazement, when he saw Borah on a bridle path, that the horse and the rider were going in the same direction.) This is not entirely accurate--he was firmly opposed to the League of Nations and the World Court, but he did advocate some positive ideas (whether good or bad), notably disarmament, the "outlawry of war," and the recognition of the Soviet Union. Since the American people weren't willing to spend much on armaments, anyway, at that time, and since the Kellogg-Briand pact was ratified in OTL (largely at Borah's urging) there are only two areas where I can see him making a difference in foreign policy: first, in recognizing the USSR, and second--and probably most important, since I don't think earlier US recognition of the USSR would have more than symbolic effect--in opposing US military intervention in Latin America, notably Nicaragua. "President Coolidge had his Senator Fulbright in the liberal Republican William E. Borah of Idaho, who kept demanding to know the true casualty figures of the U.S. force and of those opposed to it."
http://www.americanheritage.com/content/mr-coolidge’s-jungle-war
Would Borah have been able to secure the Republican Party's nomination for the presidency in 1928? It's hard to say. He certainly had a lot of enemies in the party, and his reputation as a loner would not help him--a commentator on the 1930 Congress recognized "four distinct political factions" at work: "Republicans, Democrats, Progressives, and William Edgar Borah."
http://www.kevincmurphy.com/williamborah.htm But OTOH it is not all that easy to deny an incumbent Republican president renomination--if nothing else, there are all those patronage-dependent delegates from the South (who came in so handy for Taft in 1912). His opposition would be divided between Hoover, Dawes, Lowden, and others, each of whom would have his own weaknesses and enemies. And maybe the Old Guard would remember that whatever Borah's heresies, he always faithfully supported the national ticket in every presidential election--even in 1912 he backed Taft, whereas Hoover was a Bull Mooser (and moreover would in effect support Wilson's call for a Democratic Congress in 1918).
Anyway, if we get Borah for four more years, his approach to the Depression is likely to be considerably different from Hoover's (though also different from the New Deal). "Citing that '4% of the people of the United States own 80% if its wealth,' Borah advocated the lowering of tariffs to promote world trade, inflationary monetary measures such as paying depositors to put their money in circulation and reissuing silver, and, most importantly, the destruction of monopolies, which were 'bleeding our people white.'"
http://www.kevincmurphy.com/williamborah.htm But while advocating reflation and redistribution, he "found the dubious constitutionality and extralegal bureaucracy of several First New Deal measures to be as large a threat to American citizenry as monopoly itself. 'You can not exchange the regimentation of government for the regimentation of monopolists,' he declared." Ibid.
Anyway, it's hard for me to see Borah winning re-election in 1932. Even if we assume that he could get his anti-monopoly and reflationist legislation through Congress (which I doubt) I do not see them curing the Depression. It might not be quite as severe as in OTL but people in 1932 are going to be looking at how bad things were compared to 1928, rather than imagine that they might have been even worse under another president...