"It's hard not to think, 'What if he was here and what if he was present for this breakthrough?' " says Reeve, who offers PEOPLE an exclusive first look at a video showing the amazing progress made by four young men paralyzed by spinal cord injuries.
The men were told they would never be able to move again below their neck or chest but are now able to stand and move their hips, legs and toes with the help of a new therapy called epidural stimulation.
"As much as we miss him," the 34-year-old writer and director says of his famous dad, "there's a comfort in the fact that we wouldn't be where we are had it not been for his tireless advocacy."
Indeed, after the Superman actor was paralyzed in a 1995 horseback riding accident, he spent the last years of his life lobbying for cutting-edge research that might offer a cure. He and wife Dana Reeve, who died of lung cancer in 2006, worked tirelessly to improve the lives of six million Americans living with paralysis through the Christopher & Dana Reeve Foundation.
Just a teenager when his very fit, athletic father became a quadriplegic, Reeve recalls how the latest research was often the subject of dinner table discussions.
"It was a big part of all of our lives," Reeve says of himself and his siblings, Alexandra Reeve Givens, 30, a senior counsel to the Senate Judiciary Committee, and Will, 22, a recent college graduate. All are involved in supporting the foundation's work.
"It's a huge honor to continue the work he started," Reeve says. "I don't come close to being as effective as he was, in terms of raising money and increasing awareness and his advocacy efforts, but it's a cause that's close to my heart."
Ken Regan / LFI
The treatment involves implanting a device on the spine normally used to treat pain to stimulate the nerves and remind them how to work again.
"This is absolutely life-changing," says Rob Summers, 28, a former college baseball player and now a coach and motivational speaker.
"It's given me self-confidence," says Dustin Shillcox, 30.
Not only have the men in the study regained the ability to stand and move below their waist, but the treatment has also had an unexpected benefit for all of them: the return of bowel and bladder control and sexual function and improved cardiovascular and respiratory function.
"With the stimulator, it feels like I'm normal again," says Andrew Meas, 35, a husband and father who was among the four who received the implant.
Kent Stephenson, 27, who was told he would never be able to move from his chest down, says in the video, "I can't wait to skip, leap and jump again."
Calling the new treatment a "game changer," Reeve says, "It's no longer a question of if there will be effective treatments for spinal cord injury. It's a question of how quickly we can get them to the people that need it most."
"My father dreamed of a world with empty wheelchairs and gave hope to a whole community," he says. "This is a key new step in that hope becoming realized."
He hopes the stories of the men featured in the video will inspire others to support the research.
Watch the video below. To learn more about the BIG IDEA campaign, go to ReeveBigIdea.