The 39-year-old firefighter died a short time later at a hospital.
"That's what they were there for, was to catch a ball," Shannon Stone's mother, Suzann, said. "Cooper loves baseball, and he's a big Josh Hamilton fan. Had his jersey."
Pitching great Nolan Ryan, now the team's president, said the tragedy "hits us at our roots of who we are."
"We're about making memories, family entertainment," he said. "I certainly understand - and I'm no different than our fan base - when I was younger and I went to the ballpark, my hope was to get a foul ball.
"You can see how many people come into our ballpark with gloves, just hoping to have that opportunity," Ryan said. "That's just part of the experience of being there."
On Friday, players had the option of getting grief counseling; both teams planned to wear black ribbons on their uniforms. At Rangers Ballpark, flags flew at half-staff, and a black tarpaulin covered the gap where Stone fell.
A moment of silence was planned before the Rangers and Oakland Athletics played the second game of their four-game series.
Hamilton, still grappling with the aftermath of the wrenching night, said Friday he could hear the boy screaming for his dad after Stone fell. The player said he remembers the fall "like it happened in slow motion."
Jenny Stone, the victim's 36-year-old widow, worried how her only son would recover from the horror of not just watching his father fall but riding in the front of the ambulance on the way to the hospital.
At the request of the Stone family, MLB.com has not posted video of the accident.
Replays showed the boy watching as his father stretched and reached out to grab the ball and then fell through a gap of several feet between the left-field seats and the 14-foot-high outfield wall that has a video scoreboard on it.
City officials said the building code requires the guardrails to be at least 26 inches high. Ed Dryden, Arlington's building official, said railings throughout the park are 33 inches high.
There have been other falls at the 17-year-old stadium. Last July, a man survived after tumbling from an upper deck as he tried to catch a foul ball. In 1994, a woman fell about 35 feet as she posed for a picture after the Rangers' first game.
After last year's accident, Ryan said the team studied the railings and felt safety was adequate; he said he wasn't prepared to say if any changes might be made now.
"As an organization we are going to be looking into this because our No. 1 concern is the safety of our fans," Ryan said. "We'll do whatever we have to do to make this stadium as safe as we possibly can for our fans."
Major League Baseball promised a review of the incident "to ensure a safe environment for our fans."
It was the second fatal fall at a major-league stadium this season.
In May, a fan died after falling about 20 feet and striking his head on concrete during a Colorado Rockies game; witnesses told police he had been trying to slide down a staircase railing and lost his balance.
John McHale Jr., Major League Baseball's executive vice president of administration, said there is no centralized process for overseeing safety at ballparks and the Texas accident may change that. He said most safety issues are left to the clubs.
