Its not so unusual for a 51 year old man in 1860 to die a natural death suddenly,
suppose Lincoln does? Who gets the Republican nomination? Who's elected POTUS? How does all this influence the next four or five years?
Let's see. Chase is too radical. Bates is too conservative. Cameron has little support outside Pennsylvania. Seward is the obvious candidate, but has a reputation for radicalism because of the "higher law" and "irrepressible conflict" speeches--though the latter was no more radical than Lincoln's "house divided" speech. Seward's reputation for radicalism was always exaggerated, and he tried to rebrand himself as a moderate after Harpers Ferry, but this simply displeased both sides. Moreover, Seward was disliked by the nativist element in the Republican Party. In spite of these drawbacks, Seward might well get the nomination without Lincoln around. But if he doesn't--how about Lincoln's fellow-Illinoisan Lyman Trumbull?
IMO either Seward or Trumbull will win in November. The only electoral votes Lincoln won really narrowly were those of California (4 electoral votes), Illinois (11), Oregon (3), and New Jersey (4 of the state's 7). It is possible that Seward would lose all these twenty-two votes, but that would still leave him with 158--six more than the 152 necessary for a majority.
http://psephos.adam-carr.net/countries/u/usa/pres/1860.txt Indiana is also sometimes listed as one of the states a Republican other than Lincoln might have lost, but in fact he won it by 8.7 percentage points, and I doubt that other Republican candidates would have done that much worse. What made Indian a very difficult state for the Democrats was the fact that Senator Jesse Bright hated Douglas and supported Breckrinidge--with the result that Breckinridge did substantially better in the state (4.5 percent) than in the rest of the Old Northwest. I also don't see any plausible Republican candidate losing New York, which went to Lincoln by 7.4 percentage points.
It has sometimes been argued that President-Elect Seward would have been more open to compromise than Lincoln, but no compromise would have prevented South Carolina from seceding and the only compromise the secessionists outside South Carolina would have accepted IMO was Crittenden's, with its guarantee of slavery in all territories south of the Missouri Compromise line including those "hereafter acquired"--which could mean anything from the Rio Grande to Tierra del Fuego. (And this was in an era when further US expansion into Mexico, Central America, Cuba, etc. was widely expected.) This would be going too far even for a relatively moderate Republican like Seward. (As for Trumbull, he would have been as firm as Lincoln in resisting any extension of slavery.)
Seward would however have been more likely than Lincoln to give his approval to some of the compromise proposals then made, e.g., the so-called "border state plan." The border state plan was a milder version of the Crittenden Compromise. It was designed to be more palatable to moderate Republicans. Unlike the Crittenden Compromise, it only restored the Missouri Compromise Line in existing territories, and did not apply it to territories "hereafter acquired"--those two words seemed to Republicans to be an invitation to slavery expansion into Latin America. Also, the border state plan did not use the word "slavery", which some Republicans did not want to put in the Constitution. The border state plan would never have satisfied hard-core secessionists, but might have swayed wavering voters in some other Deep South states, and have given victory to "Cooperationists" [1] over "Immediate Secessionists in states like Georgia and Louisiana. This would at least slow down the formation of the CSA but whether it would prevent it is IMO doubtful.
[1] To quote an old soc.history.what-if post of mine:
"As it was, the victories of the 'immediate secessionists' in the elections for secession conventions in the Deep South states were quite narrow. Analyzing the results is difficult, because in most cases the competition was not between secessionists and outright unionists but between 'immediate secessionists' (through separate state action) and 'cooperationists.' The cooperationists said that they too wanted to secede, but only after a southern convention, which (according to some of them) might present the North with the South's final demands for preserving the Union. Historians are divided on the extent to which cooperationism was just another form of secessionism or a disguised form of unionism. At the very least, cooperationism would have slowed down the momentum of secession. And to some immediate secessionists, the cooperationists were requiring such a degree of southern unanimity as a condition for secession that disunion would be very unlikely.
"In any event, even in the Deep South, the contest (in December 1860 and January 1861) between immediate secessionism and cooperationism was often very narrowly decided. In Alabama, the secessionists cast 35,600 votes, the cooperationists 28,100. In Georgia, the secessionists won by only (at most) 44,152 to 41,632. In Louisiana, the secessionists prevailed by 20,214 to 18,451. In Mississippi, there were 16,800 votes for secessionists, 12,218 for cooperationists, 12,000 for candidates whose position was not specified or is now unknown. Florida was somewhat more pro-secessionist than, say, Georgia, but even in Florida the cooperationists got about 40 percent of the vote. (My source for these figures is David Potter, _The Impending Crisis_.)"