Lee brought kung fu to the Western mainstream in the 1970s, becoming a global movie star before his untimely death at the age of 32.
The popularity of his groundbreaking fusion of fighting styles pre-dated the current boom in mixed martial arts (MMA) contests by four decades, and helped the likes of Chuck Norris become Hollywood action stars.
"It has been 40 years since he passed and everybody still knows his name. We've got people from all across the world watching One Championship looking for the next big star.
"Asia hasn't had a lot of homegrown sporting heroes apart from Manny Pacquiao or Yao Ming -- but we can have more," added Cui, referring to the Filipino boxer and the Chinese basketball star.
According to Cui, every great martial artist in the world has come from Asia. Along with Lee, he lists Jackie Chan and Jet Li, who also transcended their Asian fame to star in Hollywood films.
"People in Asia get MMA -- it's in their DNA," Cui says.
"You're not having to start from scratch, people understand it. It's part of their lives -- every country in Asia has their own martial art, whether it be karate, muay thai, kung fu, silat, or jiu-jitsu."
One of the fastest growing stars in the One Championship stable is Angela Lee
, who is making waves in the world of MMA.
Like Bruce Lee, she has a mix of American and Asian background.
Born in Hawaii but of Singaporean descent, the 19-year-old has won all three of her professional bouts -- her next is in the Philippines on December 11.
Her father Ken, who acts as her full-time coach, and mother Jewelz are both MMA champions in their own right, while her brother Christian will make his professional debut in Manila on the same card as Angela after enjoying success in junior events.
Her other two younger siblings are both in training but their experience will be vastly different to hers.
Growing up, Lee was often the only girl in the class -- not that it bothered her at all.
"I had to fight against boys all the time -- it was fun, I enjoyed it," she said.
"I was really competitive when I was young. I did really well. I was really strong for my age.
"At first, the boys didn't want to fight me. I'd end up beating them and they'd end up being upset -- it was kind of funny.
"It gave me confidence. I told myself, 'I'm fighting against these boys all the time and it's going to make it that much easier when I face these women when I face these women in the cage.'"
Lee's emergence has come at an opportune time given the rising profile of women's MMA, helped by the success of American fighter Ronda Rousey
and the UFC.
Rousey, sensationally defeated by Holly Holm in front of a crowd of 56,000 in Melbourne last month, has helped revolutionize the sport.
Her antics in the cage have attracted not only sponsorship deals but also a whole new audience -- and a new generation of young fighters.
"Times are changing now," said Lee, who turned professional in May.
"It's better with all of these positive role models and that's what I hope to become for the younger girls, to encourage them to take part in martial arts.
"It is growing at a rapid rate. In Asia, they're really expanding their horizons for women's MMA. I want to be a part of that.
"I really love fight night. All those fans, all that energy, it makes me so happy and I'm really looking forward to it."
Some 20,000 fans are expected to pack out the Mall of Asia Arena in Manila, where some of the biggest names in the sport will be on show.
"The One: Spirit of Champions" event will host the franchise's inaugural men's heavyweight world championship plus five other bouts.
"Our TV deals means that we're shown on free-to-air TV in a lot of countries," says Cui.
"People can watch and see the best Asian MMA stars in the world.
"It means we can reach more people, attract more people to the sport and perhaps get some involved.
"It's unlikely you're ever going to get a Cambodian golf champion or Singaporean tennis champion, but you will get MMA stars from those countries."
Having worked in marketing around some of the biggest sporting events in the world such as the Olympics and golf's PGA Tour, Cui's ambitions are global.
UFC may be the dominant force in the West but Cui believes there is also room for One Championship.
"UFC has been in the game for a long time," says Cui. "It doesn't matter what business you are in, there's always the West against the East.
"It's Apple against Samsung, Twitter versus Weibo, Amazon and Alibaba, Ford and Kia," he added referring to some of the biggest brands of the Western and Asian worlds.
"We've got two billion people in the same time zone, so we are working completely differently.
"UFC is the biggest in the West -- One Championship is the biggest in the East."