"Professional Wrestling" itself is arguably this since anything about it that resembled the traditional Greco-Roman style wrestling all but died out by the early 20th century. Perhaps in response to this, Vince McMahon coined the term "sports-entertainment" towards the end of the millennium, but most fans refuse to use it as they consider it a Forced Meme.
The "Martial-Arts" part in Frontier Martial-Arts Wrestling was based in Atsushi Onita's original idea of creating a shoot-style promotion like the Universal Wrestling Federation, in which he was barred from entering. This remained only as a name, as the company only went a couple shows with martial arts matches, which were more in the vein of Antonio Inoki's "mixed style fights" before moving onto the hardcore matches it would become best known for, and the martial artist they gave the most recognition to, Masashi Aoyagi, isn't even that well known among the fandom.
Advisory and Management Assistance. It was thought up during the conception stages of an umbrella company that would include several touring businesses. The lucha libre group was the only one to get off the ground but the name "AAA" stuck.
In 1995, Jonathan Rechner debuted an evil Santa Claus gimmick called "Xanta Claus." His finisher was a Michinoku Driver variation called the Nutcracker Suite, a name that would stick long after he became Balls Mahoney. Then again, "nutcracker" can also refer to, well...
After Shawn Michaelsturned heel on Marty Jannetty on Brutus "The Barber" Beefcake's "The Barber Shop" on the January 12 (taped December 3, 1991), 1992, WWF Wrestling Challenge, he started using "The Heartbreak Kid" as his Red Baron, and his theme song was titled "Sexy Boy." While Shawn remained good-looking, he moved past his old gimmick into that of the "Showstopper" during his World Title reigns four years later.
Triple H's Finishing Move, the Pedigree. It's a reference to his time as snotty BluebloodHeel Hunter Hearst Helmsley, a gimmick that he has long since abandoned. However, it is still occasionally referenced to whenever someone calls him "Hunter" instead of Triple H.
Faarooq and Bradshaw's tag team, the "Acolyte Protection Agency", takes the "Acolyte" part from when they were members of The Undertaker's "Ministry of Darkness" stable. However after the end of that storyline they changed their gimmick to beer-drinking guns-for-hire with an "office" that only consisted of a free-standing door back stage and a card table, keeping the "Acolyte" part from their days as minions of a demonic overlord (who subsequently became a biker...go figure).
In that regard The Undertaker himself stopped being an "undertaker" after the mid-90s. Particularly during his stint as the leader of the Ministry of Darkness, which cast him as a cult leader that was simply obsessed with death and evil forces. The biker thing is even further out than that but at least "The Undertaker" is a fairly intimidating nickname for a Badass Biker.
The "CM" part of CM Punk's name. It first stood for "Chick Magnet" (appropriately enough) from a Tag Team he was a part of, but has long since lost its original meaning. It now means whatever the hell Punk wants it to mean at the time.
Chris Jericho debuted in WWE saying that he was the "Y2J Problem", playing off of the very 1999-specific reference to the Y2K bug, yet more than a decade and a half later, with the technological relevance of the reference long, long since passed, the Y2J nickname has stuck all these years.
Arena Femenil was so named because it was the base of all women's fed LLF that was founded in 2000, but while that remained the case, many other feds have since run shows in the facility, Nueva Generación Xtrema doing an all male show as early as 2004.
The H in ROH became an artifact after the code for which it was named was suspended. Even when the code of honor was reinstated, it was done so as an optional guideline rather than the law of the promotion.
Total Nonstop Action was never intended to be such, rather what those initials more commonly stand for was much more important to TNA. It ran into this trope when it got it's second television deal, the launching of "iMPACT" on Fox Sports Net, which saw the product become more family friendly. While iMPACT would regain the "edge" the company was originally known for with the move to Spike TV, female assets were never really used as a main selling point again.
John Cena's finisher was originally referred to as "The FU" in response to Brock Lesnar's F5 finisher. The name of The FU was changed to The Attitude Adjustment since the previous name no longer fit Cena's character. The name of Cena's trademark fist drop, the Five Knuckle Shuffle still remains intact.
SAnitY: While the nit (Nikki Cross), A (Alexander Wolfe), and Y (Eric Young) still work, the S was for original member Sawyer Fulton.
Certain wrestlers in the WWE Hall of Fame never wrestled for WWE, such as Abdullah the Butcher. It also includes wrestlers whose time in WWE is more or less a footnote compared to their accomplishments elsewhere (such as Sting and Diamond Dallas Page) and personalities only tangentially related to the product through brief appearances (like Drew Carey, whose only noteworthy accomplishment in WWE was eliminating himself from the 2001 Royal Rumble). The Hall of Fame has grown to be more of a shrine to the industry's figureheads more than ones that specifically contributed to WWE.
It should be noted that WWE has bought up the video libraries and intellectual property of those other promotions, such as WCW, and they are all therefore under the WWE's corporate umbrella, if not the wrestling promotion's.
"Stone Cold" Steve Austin was best known as a loudmouthed hot-head, so "Stone Cold" probably sounds like an Ironic Nickname. But when he first started using the moniker, he was a cold-blooded ruthless individual (based on "The Iceman" Richard Kuklinski) who spoke in a Creepy Monotone.