Ascended Fanon: Sephiroth's signature black wing never appeared in the original game. While he was the original One-Winged Angel, and had the black wing in his final form, he never had any wings in his human form. Instead, said wing made its first appearance in Kingdom Hearts. It seems to have proven popular with (most of) the fanbase, though, since it appeared in Advent Children and pretty much every game Seph's made a cameo in since.
Network to the Rescue: Not only did Sony help with the costs for the game, they also handled the North American distribution (Square didn't have an American distributor at the time).
No Export for You: Before Crisis, particularly annoying because it's almost universally agreed by reviewers to be the highest quality cell phone game of all time. There may yet be hope now that Square has ported two cellphone projects to other platforms explicitly to ensure Western release, but this one remains a curious omission.
The Other Darrin: Twice with Aerith, counting Mandy Moore from her Kingdom Hearts appearance. Since then she's been voiced by Mena Suvari in Advent Children and Kingdom Hearts II, and has since been replaced by Andrea Bowen for Crisis Core and Dissidia 012.
Sephiroth's VA in Kingdom Hearts was Lance Bass. George Newburn took over the role in Advent Children, and has held on to it ever since.
Yuffie's VA in Kingdom Hearts and Advent Children is Christy Carlson Romano; in Dirge of Cerberus (and Kingdom Hearts II, for that matter), she's played by Mae Whitman.
According to legend (referenced extensively online, but apparently without magazine scans/transcriptions of those interviews), Hironobu Sakaguchi's original pitch for Final Fantasy VII was a Detective Drama taking place in modern-day New York City. Obviously, these segments were incorporated into the first few hours of the game, followed by a more traditional RPG Overworld Not to Scale once Cloud leaves Midgar. Eventually, Sakaguchi had his way and got his NYC gumshoe plot made, starring a gender-flipped Cloud, to boot.
Many cut concepts for the game, including Cloud really being a Sephiroth-created copy of the original Cloud, who would occasionally speak through him.
Most interesting is the idea that "jenova" would be the term for a certain aspect of humans that when "awakened" with mako gave them magical abilities. Again, this was later incorporated into Parasite Eve with its power-hungry mitochondria.
Sephiroth's "jenova" powers would be heightened, and his madness caused by over-exposure to mako. In other words, Sephiroth could have ended up being a silver-haired Expy of Kefka.
Red XIII had an entire sub-plot based around him cut from the game. He was going to have his own Quirky Miniboss Squad of feline clones created by Hojo that would repeatedly attack the party.
Yuffie also had a major character-based subplot cut from the game due to time constraints, in which she was a former SOLDIER hired by Shinra to kill Sephiroth and Cloud. She was also intended to be 25 years old, nine years older than she turned out in the final game.
Early concepts of Vincent had him wielding a scythe instead of a handgun. His profession changed from horror researcher to chemist, to detective and finally the ex-Turk turned undeadgunslinger we know today. His detective concept had him, the party, and even the player unaware of his shapeshifting powers until some time after he would join the party. Vincent would be the most shocked upon awakening his powers - and his various forms would carry over into the Field maps and cutscenes.