Hitters at the Mendoza Line are almost definitionally not very good, and players with poor averages usually don't hit a lot of home runs. You have to hit the ball to hit it out of the yard, after all. Lately, though, there have been more exceptions to those rules than ever before.
This season, Chris Davis is in line to become the 25th player in MLB history to hit under .210 and hit 20 or more home runs. (.210 is an arbitrary mark, but we set the bar there to catch those whose flirtation with the Mendoza Line has turned to outright romance.) Mark Reynolds and Mike Moustakas are on pace to join the list as well. If all three maintain their averages and home run rates, this season will tie 2010 for having the most players to reach this "achievement."
This is an interesting thing, an expression of the modern offense. Despite a lot of handwaving about steroids and shifts and such, the real difference between baseball as it's played now and as it was played 10 or 15 years ago is just that hitters are striking out a lot more; when they actually make contact, they're hitting basically the way they did when every team had a chemistry lab set up in the clubhouse. More strikeouts lead to lower batting averages, meaning there are more sub-.210 hitters, but because balls are still flying when they are hit, some shitty hitters are going to show off some real power to go with their terrible averages.
The reason there were only three cases of this before the '80s is the same reason homers in general weren't as common—prioritization. It's common for hitters today to be willing to trade contact for power in a way that was pretty rare in the past. Consequently, of the 25 players to have this home run-batting average combo, eight of them (nearly one-third) have come since 2007, with a few more likely to be added by the end of the season.
Here's a table showing every player who batted under .210 with more than 20 home runs. As you'll note, it's not the worst list to be on. You have to be pretty good at something to keep a job while being unable to make contact with a baseball.