On Thursday, TMZ Sports reported Beckham was released on $5,000 bail after being booked on counts of "unlawful carrying weapon" and "possession of marijuana under two ounces."
No further information about what led to the arrest was immediately released.
The elder Beckham played college football at LSU from 1990 through 1992. The running back rushed for 757 yards and seven touchdowns on 152 carries and added 30 catches for 292 yards.
His son followed in his footsteps by playing for the Tigers in college. He was named a First-Team All-American in 2013 before being selected with the 12th overall pick by the Giants in the 2014 NFL draft.
Beckham is going to miss the rest of the 2017 season after undergoing ankle surgery in early October.