Beam Me Up, Scotty!: 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.
More or less. "Terra's Theme" IS Terra's theme, it plays when she rejoins the party in Mobliz and part of it is mixed with "Omen" the opening theme. Both of these are the times when she could be considered to have joined the party. "Awakening" is the theme that plays when she has an epiphany and even then the melody is like that of "Terra's Theme". So "Terra's Theme" is there in one way or another whenever something Terra-related happens.
Deleted Scene: The game was one of the first 24mb cartridges for the SNES but several events still had to be cut due to space limitations. They are known through sources like old V-Jump articles.
Most famous is the exchange between Strago and Shadow in a bar. Strago tells Shadow he wants to see his face for Relm's sake, to which the latter unmasks with his back to the camera. The scene is also illustrated in a Carddass.
If time ran out in the damaged building in Tzen, instead of a standard Game Over, Sabin was planned to be caught in the collapse and die. Bringing Edgar to the site would make him spend all night trying to dig out his brother, even though too many in-game days have passed after the fact. This was altered partly due to being too dark.
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.
Take a Third Option: The Genji Glove you can get in the Returner Hideout is far more useful than the Gauntlet you can get as an alternative, but getting the Genji Glove requires you to continually refuse Banon and miss some important plot development. However, if you refuse Banon once and then talk to the Returner walking around in the treasure room where you can get the Air Lancet and the White Cape, he will give you the Genji Glove. You can then go and accept Banon's request, and watch the Returners' strategy meeting.
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.
Parts of this concept ended up inspiring the characters Tidus and Zidane, and in concept art this character heavily resembles Squall Leonheart.
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.
Word of Saint Paul: Soraya Saga was one of the scenario writers of the game and was responsible for creating Edgar and Sabin. She wrote a doujinshi that detailed the background history of the Figaro family such as the death of King Figaro and Edgar's mother issues. She also wrote that the reason Sabin, Gau and Cyan got along so well was because they found a surrogate family with each other. However, Square has not confirmed or denied how much of that is considered canon and Saga even said it's just a side story.