When Terry Sloane became the original Mr. Terrific in 1942, he emblazoned his uniform with the words "Fair Play" because he believed in playing fair. Sometimes it's not that complicated.
Okay, so there's a little more to it than that. If you ever go back and read Mr. Terrific's first appearance, you might be surprised to find out how much of that story is built around Sloane wanting to kill himself. Born with a photographic memory, a brilliant mind, and physical gifts that made him a star athlete, Sloane accomplished everything a person could be asked to do by the time he was 20, and was left with the feeling that life held nothing more for him. Before he could throw himself off a bridge to experience death, however, he wound up saving the life of someone who had fallen, and realized he could put his talents to use by helping people. Even with that newfound heroic attitude, though, Sloane kept the guilt of feeling like he'd been born with advantages that other people simply didn't have.
Sloane's motto in turn inspired the Modern Age Mr. Terrific, Michael Holt, but with a twist. Like his predecessor, Holt was brilliant and physically gifted—a technologist, Olympic gold medalist, and martial artist whose abilities rivaled Bruce Wayne's—and while he was similarly suicidal, his feeling was that life hadn't been fair to him. For all his talents, he couldn't prevent his wife and child from dying in a car accident. By taking up Sloane's mission himself, he was dedicated to preventing those unfair tragedies from happening to others—hence keeping the "Fair Play" logo as part of his costume.
It's worth noting that the Modern Age Mr. Terrific costume can be seen on TV's Arrow in a form that's pretty accurate to the comics—"Fair Play" and all. On that show, though, the explanation is that Curtis Holt has adopted the jacket and name of his favorite pro wrestler, a babyface who was so good that he never had to swing a steel chair to win a match.