I consulted Wikipedia:
 Dugout choice in MLB
Which team occupies the dugout on the first-base side or the third-base side is purely arbitrary. The Major League Baseball Rulebook is silent on the subject. There are many anecdotal reasons why one dugout is chosen over the other. One is that in the early days of the game the manager also served as the third base coach, so occupying the third base dugout meant less walking for the manager between innings. Contrarily, the thought is that since more close plays occur at first base than third, the first base dugout is preferred. However, the most likely theory is simply that the home team chooses the better clubhouse and the dugout on that side of the field. (For example, prior to their 2008 move to Nationals Park, the Washington Nationals occupied the third-base dugout at RFK Stadium because it was the larger and newer of the two dugouts.)
In the National League, far more are on the first-base side (11 to 5). In the American League, though, it's split evenly, with seven on each side of the field. Even the two oldest parks still in use differ on this point: the Cubs sit on the third-base side at Wrigley while the Red Sox inhabit the first-base dugout at Fenway. However, the four parks that have opened most recently (those in Cincinnati, San Diego, Philadelphia, and Washington) all have the home dugouts on the first-base side.