Benefits of Retaining the Electoral College
It respects the Great Compromise between the states - The Great Compromise, meant to protect small states from getting overwhelmed by the larger ones, necessary to get them to agree to form the Union, extended not just to the legislative branch but the executive as well. By adding the equal number of senators to the number of congressmen to determine each state's electoral votes, small states were given additional weighting in the say they would have in choosing the President.
For example, one state A with twice as much population as B might have 6 representatives to B's 3. However, the number of electoral votes is not 6 to 3 or twice as much say but 8 to 5 due to the addition of the senators, resulting in just 1.6 times as much say, honoring the compromise. Switching to a popular vote would eliminate that weighing benefit, as state A would have roughly twice as many voters as B and hence twice as much say.
This benefit given to the small states has shrunk by about a third since the country was formed. In the First Congress there were 68 congressmen and 26 senators generating a total of 94 electoral votes, so that a state's population was only a 68/94 = 72.3% factor of its say. Since the number of congressmen was fixed at 435, population now provides 435 / 535 (leaving out D.C.'s 3 electoral votes which aren't based on representation) or 81.3% of the weighting.
States are better protected from voter irregularities occurring in other states - With the Electoral College, any ballot-box stuffing done by one state does not limit the say of others. Take one state with say 10 electoral votes, for example, they could have three, five, ten million falsified ballots for a candidate, they can go to town, it wouldn't matter, only that state's 10 electoral votes would be corrupted. Pennsylvania (20 votes), Iowa (6), Arizona (11), etc., would still have their say and their respective weighting. If we switched to a popular vote, those invalid votes would count to the national total, harming the say of states that run their elections more cleanly. In general:
With a national vote election, states which do a better job ensuring valid voting (photo ID, checking to make sure voter registrants are US citizens, alive, etc.) would lose say over states that tend to look the other way in such matters.
Conversely, because we use an Electoral College, there is much less incentive for ballot-box stuffing as it doesn't buy one party-dominant states tempted to engage in such activities anything. It results in cleaner elections, and hence a cleaner society, nationwide.
A national vote would encourage a race to the bottom in granting suffrage ("State A is now allowing their green card holders to vote, so we should also lest we lose our say...", "State B is now allowing 16 year olds to vote...", etc.)
Close elections are much easier to resolve - The 2000 election between Bush and Gore took about one month to resolve but the focus thankfully was just with Florida and its then-25 electoral votes. Because the Electoral College was all that mattered, the other 49 states could be ignored, 20 inaccurate votes in Idaho for example or 30 missed votes in California could be disregarded. However, if we had a tightly close popular vote election, then every precinct in every county in every state would be subject to re-examination, to grab 5 votes here and 10 votes there because they all count towards the national vote total, not a situation anyone would relish.
With the Electoral College, people are more likely to vote for a candidate based on his/her policies rather than on whether or not he/she has visited their state. - Voters in places like California and Tennessee where the outcome of the vote is not in doubt are perfectly fine with their candidate of choice never visiting their home state during the campaign. More than that, they don't actually want their candidate wasting time in their home states, instead wanting him or her to be in battleground states trying to win them over so their candidate can win. This provides a lot of relief to presidential candidates as it allows them to focus on visiting just 10-15 battleground states, a grueling enough task as it is.
With a national popular vote, however, it would still make sense to visit vote-rich states regardless of their leanings, causing many voters to get offended if their candidate does not visit them, and possibly switch votes as a result. So the Electoral College is more likely to result in voters casting votes based on the candidate's policies than whether they visited their state, good for the nation as a whole.
The Electoral College makes election night a lot more exciting to watch - A silly argument, but still true. With the Electoral College, viewers see a scoreboard of states that slowly get lit up in red and blue during the evening, for which viewers can celebrate or mourn each time a candidate wins or loses a state. With a national vote you're just waiting for a vote total at the very end, rather dull for all concerned. It would be similar to watching a football game in which all touchdowns and field goals are irrelevant but just the total yards gained within the 60 minutes of play determining the victor.
Finally, it should always be noted that a candidate who wins the popular vote and loses the electoral votes on an election based on the latter wouldn't necessarily have won the popular vote if it were based on the former, as the campaigning done--states visited, TV advertising strategy, issues chosen, voter turnout generated, etc.--is different under the two styles. A skilled candidate who can win with one set of rules quite frequently has the ability to win under another.