I had a post about this some years back in soc.history.what-if:
James Garfield, nominated for president by the Republican Party in 1880, knew
that to win the election, he had to placate the defeated "Stalwart" faction
of the party (which had backed Grant). So he offered Levi Morton, a New
Yorker and close associate of Stalwart leader Roscoe Conkling, the vice-
presidential nomination. On the advice of Conkling and Governor George S.
Boutwell of Massachusetts, though, Morton declined, and Garfield turned to
Chester Arthur instead, who against Conkling's advice decided to accept the
nomination. (*The Nation*, which thought the GOP "could hardly have made a
better choice" for head of the ticket, had a lower opinion of his running
mate, but observed "There is no place in which his powers of mischief will be
so small as in the Vice-Presidency, and it will remove him during a great
part of the year from his own field of activity. It is true that General
Garfield, if elected, may die during his term of office, but this is too
unlikely a contingency to be worth making extraordinary provision for.")
Let's say Morton accepts. I assume that Garfield-Morton wins the same kind
of narrow victory that Garfield-Arthur did in OTL. (If anything, maybe the
victory will be slightly greater, since Morton was less notorious than
Arthur; he had been a successful businessman long beofre he became a
politician. In any event, whoever would be on the GOP ticket, Tammany boss
"Honest John" Kelly does not seem to have been too anxious to help Hancock
win the White House; he hurt the national Democratic ticket in New York by
backing a very controversial nominee for Mayor of New York City.) The
"disappointed office seeker", crazy Charles Guiteau, kills Garfield as in
OTL, though he utters slightly different words ("I am a Stalwart, and Levi
Morton is president now!") How does a Morton administration differ from an
Arthur one?
There is going to be civil service reform, in any event. To me, the real
question is whether Morton would have a better chance than Arthur of winning
re-nomination in 1884, and if so, whether he would have won election to a
full term. Obviously, he would have the advantage over Arthur of better
health. And any Republican president of that era would start out with an
advantage at a GOP convention--the patronage-dependent Southern delegates.
That got even Arthur a respectable number of delegates, and he wasn't an
active candidate for re-election. Still, to get a majority, Morton would
have to beat Blaine, who undoubtedly had more of a personal following than
any other Republican. In OTL, Morton did a pretty good job of satisfying all
the major factions in the party, with the result that he received the vice-
presidential nomination in 1888, and was subsequently elected. But such a
balancing act is a lot more difficult when you are president. Both E.L.
Godkin and Carl Schurz observed that Arthur had tried to conciliate both the
bosses and the reformers, and in the end had pleased neither, and I can see
that also happening to President Morton. But if he did manage to defeat
Blaine, I believe he would win the general election, because he would
probably do better than Blaine in New York.
https://groups.google.com/d/msg/soc.history.what-if/u_YU6stJjyU/lZ1D4XWZ6pwJ
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Also, as Rich Rostrom pointed out in a reply to my post, Morton had frontline experience in electoral politics, having been elected to Congress in 1878 on his second try. (He was re-elected in 1880, and then resigned to become Minister to France.) Arthur's only experience in public office had been as holder of a patronage position. (I am not certain, though, that this makes Morton more likely to win the nomination in 1884 than Arthur was--after all, a knowledge of how to use patronage can be handy in winning delegates. Morton's real advantage would be that he would be healthier than Arthur and presumably a more active candidate. Incidentally, technically speaking I erred in my post when I used the phrase "whether Morton would have a better chance than Arthur of winning re-nomination in 1884"; since neither of them had previously been nominated *for president* I should have said nomination, not re-nomination.)