The hints of irreverence can be traced to a private meeting in Coral Gables, Fla., with the soon-to-be presidential candidate and his rapper acquaintance.

Jeb Bush had grown fond of Pitbull, the Miami performer gone global, who seemed to share his zeal for education policy. But Mr. Bush, a former Florida governor, had a question: Why the stage name? The artist replied that a friend had suggested it years ago while they were en route to a pit-bull fight.

“Well,” Mr. Bush replied at their meeting early this year, “good thing you weren’t on the way to a cockfight.”

As his campaign has struggled to rejuvenate a languishing bid, some close to him have suggested the existence of a (very) secret weapon: It is at least possible that Jeb Bush is funny.