Well if Robert showed any interest in the office it could’ve happened. He was mentioned as a candidate from 1880-1912 and refused all of the times. The most likely scenario though where he becomes President is in 1884, he was Secretary of War and would’ve undoubtedly had President Arthur’s support.
Whether he was right or wrong, Robert Todd Lincoln was probably sincere when he said, "My father was a great man, but I am not." https://books.google.com/books?id=tPqgC3RS-7sC&pg=PA421 His reluctance to seek the presidency seems to have been genuine, not a pose. Still, in a post some years ago, I did suggest one scenario in which he could have become POTUS:
There are various scenarios under which Robert Todd Lincoln could have become Republican nominee for president, but they all founder on his disinclination to seek the job. However, here I would like to discuss a different way Robert Lincoln could have become president--and perhaps a more plausible way. That is by getting the *vice*-presidential nomination in 1884. My source is Jason Emerson, *Giant in the Shadows: The Life of Robert T. Lincoln* (Carbondale and Edwardsville: Southern Illinois University Press 2012); all quotes in this post, unless otherwise indicated, are from Emerson's book.
Lincoln's position in 1884 was that he did not want either the presidential or vice-presidential nomination and that he hoped "that no such responsibility will be thrust upon me." http://books.google.com/books?id=tPqgC3RS-7sC&pg=PA256 But of course these words imply that the nomination *might* be thrust upon him, and a widespread view in the Gilded Age was that an honorable man, though he might not campaign for the nomination, could not refuse his party's call if it chose him for the presidency. (Some applied this reasoning to other offices as well. When former Illinois Governor Richard Oglesby announced in 1887 that he would not be a candidate for the US Senate, a constituent wrote him that any man who decided for himself that he would not be a candidate was "not merely impertinent, but unpatriotic...Whether a man announces himself as an active candidate is a matter of taste....But...to declare he is not a candidate is to deprive the state of its lawful right of selection.")
Anyway, Lincoln supported Arthur (whom he had of course served as Secretary of War) for the presidential nomination, believing not only that Arthur had earned it, but that Arthur had a better chance of winning in November than Blaine (Lincoln thought Arthur could carry New York, and doubted that Blaine could). However, Arthur had too many enemies, and Blaine as expected was nominated, Lincoln getting only a handful of votes.
Once Blaine won the nomination, the convention turned to the choice of a running mate. It was likely that the Northeasterner Blaine would be balanced by a Midwesterner, and Lincoln was the obvious man. He had been an able Secretary of War, and his name was thought to be magic, especially among African American voters. (There weren't very many of them in the North, but enough to potentially make a difference in closely contested states. And though Republican hopes were dim in the South, there was still a slight possibility that black voters--who had not yet been disfranchised to the extent they would be a couple of decades later--might help the Republicans carry one or two Upper South states. Anyway, it wasn't just blacks who revered the name of Lincoln.) Many Republicans who thought him too young (41) and inexperienced for the White House, considered him an ideal vice-presidential candidate. (This was not entirely logical, given that there had been two presidential assassinations in the past two decades, each thrusting a just-elected vice president into the presidency.) "Yet once his nomination seemed certain, he immediately telegraphed the convention and not only forbade his friends to present his name but also stated that he 'would not take' the nomination. While this seems to contradict Robert's numerous statements about civic duty, to him as to so many men of the day, the vice presidency was an empty, wasted position that served no real public interest; therefore, to refuse it really was not to compromise one's principle. In the end, the second spot went to another Illinoisan, Senator John A. Logan..."(Emerson, p. 257) http://books.google.com/books?id=tPqgC3RS-7sC&pg=PA257
Does this mean that there was no chance of Lincoln on the ticket in 1884? Not necessarily. For as Emerson has pointed out--and this is something I was previously unaware of--Lincoln did privately indicate that he would accept the vice presidential nomination under one circumstance: if Arthur were nominated for president. (Note that Lincoln was the *only* one of Garfield's cabinet that Arthur had retained.) Emerson writes (pp. 257-8):
"President Arthur was embarrassed and discouraged by the rejection of his party. After telegraphing his congratulations and support to Blaine, he ordered a carriage and disappeared from sight. History has shown, however, that the president did not seriously want the nomination. He did not 'bow out' of the contest, because to do so would raise suspicions about his health, cast doubt about his competence to handle the burdens of the presidency, and carry with it the implication of cowardice to both run on his record and to face possible defeat at the polls. But his health was the reason for his inactive candidacy. Although publicly unknown at the time, Arthur knew his Bright's disease was in an advanced stage and being aggravated by the stress of the presidency, and that if elected, he probably would not live out a second term.
"The knowledge of this fact makes it even more incredible to realize just how close Robert Lincoln actually came to being president. For, despite his disinterest in being the vice-presidential nominee, telegrams he wrote during the first day of the Republican National Convention show he was willing to accept the second place only if President Arthur were renominated. [It seems to me the correct word would be "nominated," since Arthur had never been nominated for *president.*--DT] Lincoln, believing his name might be presented to the convention for vice-presidential contention, entrusted longtime friend Norman Williams with a telegram to U.S. Senator Shelby M. Cullom from Illinois. In it, Lincoln instructed Cullom to withdraw his name from contention for vice president if it was presented. Confidentially, Robert instructed Williams to deliver the telegram to Cullom 'for his prompt action if any nomination for president is made by the Convention except President Arthur....If that nomination be made, telegraph me for further instructions, as my relations to him require me to consider my proper course.' The missive makes clear that if Arthur had been nominated for the presidency, and he asked Lincoln to join the ticket, Robert would have assented out of respect and loyalty to his chief. Since Arthur died in 1886. Vice President Lincoln would have become president.
"Had Robert's intentions been known inside the convention hall during the balloting, it is likely it could have swung sufficient votes to enable Arthur to win the nomination. Robert, however, took extra care that his telegram would not be revealed. Instead of sending it to the telegram station in the convention hall, he sent it to the Western Union main office and had Williams pick it up there, rather than have it delivered to him. And, if Arthur truly wanted to receive the nomination, he could have approached Lincoln previously and announced Honest Abe's son as his running mate should he be nominated. With Lincoln on the national ticket, the Republicans would have had a much greater chance of winning the nomination..."
Is Emerson right that if it had been known that Lincoln was willing to be Arthur's running mate, Arthur would have been nominated? I am not sure. At the first ballot of the convention, "Blaine received 344, Arthur 278, Edmunds 93, Logan 63, Sherman 30, with Joseph Roswell Hawley, Robert Todd Lincoln and William Tecumseh Sherman receiving parts of the remainder." http://en.wikipedia.org/…/1884_Republican_National_Conventi… So it is certainly conceivable that if it were known that Lincoln would be his running mate, Arthur could have gotten more votes than Blaine on the first ballot. But this would still be less than a majority, and it is significant that 60 percent of Arthur's first-ballot support in OTL came from southern delegates. http://books.google.com/books?id=PVwCtvcs3NgC&pg=PA215 (Daniel Klinghard, *The Nationalization of American Political Parties, 1880-1896*, p. 215). In other words, Arthur did poorly with delegates from Republican areas, and did best with patronage-dependent delegates from an area of the country that the Republicans had no chance of carrying. Having Lincoln as a running mate would strengthen Arthur with northern delegates, but I am not certain that it would strengthen him enough. As noted, he had plenty of enemies in the party (though of course this was also true of Blaine).
In any event, this scenario does seem to me to be the most likely one by which Robert Todd Lincoln could become president. To be sure, there was some talk of nominating him for president in 1888, and conceivably he could have gotten the nomination if he wanted it--and would probably have beaten Cleveland had he been nominated--but he made it clear that he did not want it, and again got very few votes. To get a Robert Todd Lincoln who would actively seek the presidency requires so great a transformation of his personality that I am convinced that only his feeling of loyalty to Arthur in 1884 could put him on a national ticket and thus make a Robert Lincoln presidency likely (assuming that the Arthur-Lincoln ticket wins in November).