Former ECW wrestler Balls Mahoney is an obvious example. According to his backstory, he learned to fight at a young age, fighting off bullies who picked on him because of his name. He seems to have embraced it as he got older, though, considering how Double Entendre-laced his promos were and his selection of AC/DC' s "Big Balls" as his entrance theme.
WCW gave Barry Darsow a golfer gimmick in October 1998. The character was originally going to be named "Stewart Pain," a reference to Real Life golfer Payne Stewart. This was abandoned when Stewart died in a plane crash. Instead, Darsow was renamed "Mr. Hole-In-One" Barry Darsow.
Teddy Hart, whose real name is Ted Annis, whose last name rhymes with... well, you know...
Molly Holly. It worked when she was a cute, blond, pigtailed country bumpkin from Alabama, but not after she became a brunette villain.
Her kayfabe cousin, Bob Holly, had the unfortunate name "Sparky Plugg" when he started in the WWF as a wrestling race-car driver.
In UK World of Sport wrestling: "Big Daddy" was really "Shirley Crabtree". Also, he ended up taking the name "Big Daddy" after hearing it in a song, not realizing it had sexual overtones, as opposed to "person who is a role model to kids". (Considering he had an entourage of kids following him into the ring)
When El Generico signed to WWE, he was given the name Sami Zayn. While the 'Sami' part has nothing wrong with it, his last name could be a little unfortunate. While Zayn is a common Arabic first namenote he's Syrian-Canadian, in the Western world the first person that comes to mind is Zayn Malik, one of the members of One Direction. While he could be the butt of One Direction jokes in the future, nothing serious has happened yet.
Chris Hero started his career as Wife Beater, based on the sleeveless white undershirt he wore at the time. He changed it after a women's group threatened to protest a show he was wrestling on in Wisconsin.
The late actor/wrestler Robert "Jeep" Swenson was brought in for a one-shot appearance as a member of the Alliance to End Hulkamania (a Villain Team-Up of the Four Horsemen and the Dungeon of Doom for the "Doomsday Cage Match" at WCW Uncensored 96). His character was originally going to be named the Final Solution (also Hitler's name for the Holocaust), before common sense prevailed and the name was changed to the Ultimate Solution. Of course, his main claim to fame for non-wrestling fans was his role as Bane in Batman & Robin (before Tom Hardy played the character in The Dark Knight Rises).