The 25-year-old moved to the top of the leaderboard on 11-under par courtesy of four straight birdies on the back nine, but heavy rain and darkness left him needing to finish his round on Saturday morning.
Rodgers leads fellow American Rickie Fowler by just one shot and the 28-year-old world number 10 will also have to complete his second round on Saturday having made it through to 15 holes.
First-round leader Patton Kizzire sits alongside Fowler on -10 after he followed up his career best round of 62 at the El Camaleon Golf Club on Thursday with a 70.
The 2005 OHL Classic champion Brian Gay sits tied for fourth with Brandon Harkins, having shot a 65 in the morning to move up to nine-under after 36 holes.
And, after Kevin Mitchell's first round hole-in-one on the 10th hole, Jonathan Byrd hit a 135-yard tee shot for an ace on the par-three eighth hole on Friday.