"SABIN SUPLEXES A TRAIN!" Technically, it's a piledriver.
A little known one that persists probably because the reality way too complicated: The name of Terra's theme is "Awakening", while "Terra's Theme" is the name of the game's main theme, a more orchestral arrangement of the same melody.
Lone Wolf from FFV appears as a thief being held in Figaro's prison and later takes Mog hostage. Technically, Gogo counts, too.
Moogles would be this. They first appeared in FFIII and reappeared in FFV, both of which were only given Western releases long after FFVI. They first appeared in the U.S. in Final Fantasy Adventure and later reappeared in Secret of Mana, as well.
Throw It In: Kefka's introduction cutscene barring Terra's flashback (specifically the part involving his shoes being dusted) was not in the original script, but was ad-libbed by Yoshinori Kitase in order to give players an early implication that Kefka was missing a screw or two from his head, and also because the scene itself, without it, seemed too boring to make completely normal.
The persistent rumors that General Leo can join your party (as He's Just Hiding), and the existence of the Paladin Ring. And there's the General Leo Revival theory that somehow ties into Shadow having two or three more (depending who you ask) hidden dreams. Sometimes dinosaur fighting is involved as well.
Leo can be recruited, in a sense. Gau learns "Shock" if you add the Retainer enemy to his Rage skillset.
Game Genie users found that it was entirely possible to have a party with Kefka and Gestahl in it... though they were largely non-functional and shared a character portrait with Terra. This stems from various scenes in the game where they appear in-battle with their regular character sprites (as opposed to the elaborate, motionless sprites of enemies), which necessitated writing them as some equivalent to a party character in the code. Suffice to say, this led to rumors that it was possible to have them in your party legitimately.
The above two are parodied by an infamous list of "unlockable characters" that starts at General Leo, and ends with Ryu from Street Fighter, with increasingly bizarre and impossible methods for each one.
The GBA release of the game featured censorship of the scene with Celes in prison. In the original release (worldwide), Celes was beaten by her guards while still in chains. The GBA version removed the beating and showed her unchained. After fans confirmed that this cut was in the Japanese version as well, speculation that the scene was censored due to mirroring a real life kidnapping in Japan ran rampant for years. Eventually, the game's translator, Tom Slattery, revealed the story was far more mundane. Square was targeting a CERO-A content rating in Japan for the re-release, but violence against a restrained person gets a much higher age rating under CERO. CERO didn't exist when the game was originally released, and thus the content wasn't a problem in the early 90s.
Early on, there was massive speculation as to what might be unlocked thanks to finding Cyan's "Book of Secrets" in the World of Ruin. Years later, it was finally put to rest, when it turned out that was the censors' response to what was supposed to be a Porn Stash.
According to some storyline concepts, Celes was originally going to suffer from mental instability similar to Kefka due to the Magitek process, but she would have gotten over it. In addition, she actually was intended to be a spy (albeit a conflicted one), but that ended up removed, although they reused the concept for Cait Sith in the next game. Interestingly, her "spy" plotline was kept in the story (note how much the characters talk about her being a "traitor"), but what she actually did to be called treasonous is never brought up in the story, likely the result of a last minute cut.
Gogo's recruitment was also different. Originally, they planned it so that you will meet a duplicate of your characters in random towns. To recruit him, you must bring the character he's currently disguised as. Suddenly, the Siegfried imposter makes a lot more sense.