In my opinion, the strongest argument in favor of the popular vote is the recent election between Al Gore and George W Bush. Al Gore won by half of a million popular votes all across the United States. However, he lost because he had fewer electoral college votes than George Bush. The deciding state was Florida, and it's a winner-take-all state. The margin in Florida was only a little over 500 votes. 
How Often Does the Electoral vs. Popular Vote Disagree?
This was actually the first time this had happened in over 100 years. It is fair to say that it is very rare to find a case when the popular vote and electoral college vote aren't the same. However, it can happen, has happened, and might happen again.
The Dispute Continues
This is also an example of the most disputed presidential election in the history of the U.S. In fact, a case went all the way to the Supreme Court of the United States. However, the court let the Florida count stand, and George Bush became president. Many Florida voters still pointed out irregularities in the process. The video below illustrates some of these:
1 Digital History
TLDR: I am obviously a bit biased, and I don't think this is the biggest or only problem with our elections, but yes, I think we should change to a popular vote.