LAS VEGAS—The fight to never give up continues for Diana Nyad, the famed distance swimmer and motivational speaker.

So it was fitting three years ago, the night before a 64-year-old Ms. Nyad swam 110 miles from Cuba to Florida—the first confirmed person to do it without a shark tank—that she was in a CVS drugstore talking to a Cuban man who recognized her. The man emptied his wallet to show off the symbol of his fight—a $2 bill he’d carried since he was a little boy. His grandmother had given him the faded green slip so many years before as a totem to carry with him on their family’s treacherous sail from Havana to America.

And he gave his grandmother’s emblem to Ms. Nyad. For luck in her fight.

“I swam across the ocean with that $2 bill,” Ms. Nyad solemnly told the assembled crowd at ACEP16’s opening general session Sunday.

The tale, like so much of Ms. Nyad’s life story, is about the power of symbols and never giving up. The boy’s family fought adversity upon a decrepit lifeboat and crossed what Ms. Nyad calls the Cuban graveyard that separates Havana from the Florida Keys.

Similarly, Ms. Nyad never gave up on her dream to swim that perilous pathway. It took her four failures—and a 30-year break from swimming—but in 2013, she and her team accomplished a goal that Ms. Nyad first nurtured as a young swimmer growing up in Fort Lauderdale.

“All these opportunities, all these accolades, I tell you, it’s been a tsunami these three years since the stumble up on that beach,” Ms. Nyad said. But “they haven’t come because I made it. They’ve come because I refused to give up on it.”

It’s a message she said she believes resonates with emergency physicians.

“I know I’m standing in front of a group and preaching to the choir here,” Ms. Nyad said. “I’m honored to be standing in front of you and I don’t say that to every group.”

Ms. Nyad recalled a swimming teammate once told her that the difference between winning an Olympic trial can be as thin as the half-moon edge of a pinky fingernail. Ms. Nyad realized that fighting for an extra thousandth of a thousandth of a second could be the story of her life.

“Succeed or fail…every single day of your life, do it so you can’t do it a fingernail better,” she said. “You’ll never have a regret.”

By the way, Ms. Nyad doesn’t have that $2 bill anymore. Three weeks ago, she gave it to Marc Buoniconti, a quadriplegic for 31 years and president of the Miami Project to Cure Paralysis, on his 50^th birthday.

A woman who never gives up paid it forward for a man who does the same. And in that way, Ms. Nyad keeps up her fight.