The trio stands at 9-under 131 overall and leads three other players by one shot El Camaleon Golf Course.
Argentina's Jose Coceres had a 5-under 65 and shares fourth place with Bill Haas (66) and Brian Gay (67) at 8-under 132. Ryan Armour (65), Peter Lonard (68) and John Merrick (69) are a shot further back at 133.
For Funk, a seven-time winner on the PGA TOUR who claimed his second Champions Tour victory last month, leading after 36 holes is nothing new. He's done it nine other times on the PGA TOUR and twice on the Champions Tour.
For the other co-leaders, it's a rarity: Beckman has two previous 36-hole leads on the PGA TOUR, and this is Weekley's first. Beckman wasn't confident his score would hold up.
'You know, I probably won't be leading at the end of the day, but I'll definitely have a chance,' he said.
But the lead did hold up. Beckman's round was nothing spectacular -- three birdies and 15 pars -- but it was good enough to put him tied atop a leaderboard where 67 players are better than even-par through two rounds.
Weekley's round was more up-and-down. He scattered six birdies with three bogeys after making eight birdies and a double-bogey while shooting a 64 in the first round.
The front nine is where he took a step back. After playing it at four-under par Thursday, he went just 1 under there Friday.
'That was a difference between yesterday and today, between shooting six-under and shooting 3 under,' he said. 'Because I made, what, two bogeys on the front side and both of them were just two bad swings.'
Paraguay's Fabrizio Zanotti fired a 7-under 63 to move from a tie for 73rd place into a tie for 10th place at 6-under 134. He was joined there by John Cook (66), Michael Putnam (68), Mark Brooks (65), Bob Tway (63) and Q-school champion George McNeill (70).
The cut line fell at even-par 140 with 79 players moving on to the weekend.