It prevents foreign interference in US electionsThis reason comes from The Federalist Papers, No 68:
Nothing was more to be desired than that every practicable obstacle should be opposed to cabal, intrigue, and corruption. These most deadly adversaries of republican government might naturally have been expected to make their approaches from more than one querter, but chiefly from the desire in foreign powers to gain an improper ascendant in our councils. How could they better gratify this, than by raising a creature of their own to the chief magistracy of the Union?This seems to make any allegation of foreign interference (read Russia) absurd if the reasoning behind this institution is sound. I am surprised this hasn't been brought up as a reason that any investigation into this is just silly.
It prevents an incompetent from becoming presidentThe 2016 US election was one of duelling idiots. While one may defend Hillary Clinton as being well educated, she certainly lacked the knowse to deal with the election process (I refer you to Jonathan Allen and Amie Parnes book Shattered: Inside Hillary Clinton's Doomed Campaign for documentation of her lack of political savvy, but that was pretty obvious to anyone watching the campaign). 
On the other hand, the US has been saddled by someone who appears to believe he wouldn't have won. I could get into Trump's candidacy, but this is a really bad one if this is one of the reasons for having the electoral college. I'd toss in the 2000 election as another example of the wrong person becoming president.
More reasonsI found Richard Posner's Slate article defending this anachronism. In defence of Posner, his article was written in 2012 before this past election fiasco. Posner gives the following reasons to keep this: Certainty of Outcome, Everyone’s President, Swing States, Big States, and Avoiding Run-Off Elections. I have to admit that the learned judge seems to be offering confused reasons.
Certainty of Outcome is a bad one for the learned judge to begin with since Gore won the popular vote by over 500,000 votes in 2000 and Clinton won the popular vote by nearly 3 million votes (2,868,691). I found it hard to find a graph which gave the popular vote in the 2016 election, as opposed to graphics that showed the electoral vote, since this number is so disparate it makes this argument risable.  It would seem more certain in a truly democratic society, or at least one that likes to pretend to the rest of the world how great its democracy is.
Or is that pretence a relic of the cold war? Now that democracy is no longer an issue the US can get rid of its pretending its democracy is somehow better than the rest of the world.
Everyone's president is a truly laughable assertion under Trump. How many people DIDN'T vote in the last election? Then there are people like me who voted for third parties.
The reality of the "everyone's president" argument made by Posner is silly is that he then goes on to "Swing States" and "Big States". Posner is trying to use the founder's belief that somehow the Electoral college prevents regionalism. Then he goes into the glaring examples of regionalism. It was Clinton's failures in swing states that cost her the election!
Bottom line on those three arguments: you can't claim that somehow the electoral college prevents regionalism when regionalism is what ended up costing the election of someone who won the popular vote by 48.5% (as opposed to the electoral college winner who won by 46.4% of the popular vote).
Reading Posner's article, the 2016 election points out the flaws in his arguments: the electoral college serves no point other than to be anti-democratic, which gets into "run off elections".
Those would be small prices to pay if they would be the cost of having the democracy the US has presented to the rest if the world through the last part of the 20th Century and the beginning of this one.
The problem is the electoral college is an anti-democratic institution which is an extreme danger to the electoral process. The sad part is that the travesty caused the electoral college is again being ignored. I noticed that the democrats were blaming everything except this fossil for their loss. Now, the silliness of foreign influence in US elections overlooks a reason given for this artefact.
The 2016 Presidential election has demonstrated that this institution needs to be abolished, or drastically reformed. Its existence has led to a constitutional crisis (not that the US hasn't been on the verge of one since its inception). But this one is one of proportions that can no longer be ignored.
The real bottom line here is that the US system of elections is in drastic need of an overhaul: does it take a Constitutional crisis to force this to happen?
- See also:
- Ask Factcheck: The Reason for the Electoral College
- Election 2016: The Real Reason the Electoral College Exists | Time.com
- Ignore the Mob—Long Live the Electoral College - Reason.com
 disclaimer: I voted for Jill Stein for many reasons other than just the "democratic" party running Clinton, but her choice was one of many sickeners the party gave me. The entire US election process makes me sick, but the duopoly really disgusts me.
 I knew Clinton was going to lose when the election results focused on the electoral votes as opposed to the popular votes.
 there is little difference between a republic and a democracy in modern political science. Besides, the French Revolution pretty much put paid to most of the anti-democratic v republican beliefs of the founders.
 Here is my wish list of changes to the US system of elections. Only Jill Stein and the Greens seems to be willing to mention them:
open debates run by an impartial body like the League of Women Voters, shorter election cycles, open primaries, ranked choice voting, return of the fairness doctrine and equal time rule (Trump used the lack of it to get shitloads of free publicity), campaign finance reform--if not publicly funded campaigns, easier access to the ballot for parties, reform or abolish the electoral college, end gerrymandering, handcounted paper ballots or receipts, and I am sure that is only the beginning.