Chocobo Rumors:

Silver Chocobo: Supposedly if you mate two A or S class gold chocobos, you will get a silver chocobo. This chocobo supposedly gives it the ability to swim underwater like the submarine. Supposedly you can go into the Midgar Swamp with it and fight Onyx WEAPON (the weapon which was destroyed by the Shinra Mako Cannon). Of course, this doesn't make any sense since it was destroyed by Junon Harbor.

Invisible Chocobo: This has to be the worst possible rumor! A chocobo that you can't see! You have to wander around the map, trying to mount it!

Note: There are also rumors of many other chocobos, the Red, Orange, etc.

Cheats:

Slot Cheat: You can tap the square button repeatedly on slots (such as Cait Sith's and the battle arena's) to slow them way down.

Other Rumors:

Ruby Weapon: You can hit Ruby Weapon with Hades, Bad Breath, or Stop to paralyze it.

Super-Magic: This rumor's mine. Although I haven't tested it yet (I don't have a second-level Quadra Magic), it should work. Equip W-Magic and Quadra Magic (on at least Level 2). Cast Magic that Quadra Magic is junctioned to twice to get it to attack 8 times. You could try using this with W-Summon, too. 8-Knights of the Round... Hmmm. That would be good on Emerald Weapon or Ruby!

Other WEAPONS: Their are rumors that their is an Onyx WEAPON (see Silver Chocobo). Their are also rumors of a few others... Sapphire WEAPON and such, but Onyx WEAPON is the most popular.

Sephiroth: Their are many rumors that you can somehow get Sephiroth in your party. The most believable is one requiring a Game Shark (I don't have one, and don't want one, so I can't test it...) However, some say that by doing some very weird things... such as only using Limit Breaks until Aeris is killed, and getting all of your characters up to level 99 before Aeris is killed. Another one says that if you somehow get the Masamune, he will join you. An add-on to this rumor says that Rufus will take over the Meteor storyline.

Masamune: Another rumor says that if you collect more than one piece of Mithril (most say 5) from the blind man, you can rebuild the masamune and either Sephiroth will come, or Cloud can use it. However, this seems extremely unlikely (as do most of the rest!) because I haven't been able to get more than one piece of mithril.

Cid's Automobile: In Cid's house is a broken down wreck of a car. Yet another rumor says that if you can get the Super Sweeper (which you supposedly get from the Boss Battle at the Battle Arena (I've only ever gotten autographs)), so many 1/35 Shinra attack Soldiers (usually they say 10-12... I've only ever found 2), and if you haven't used the mithril, that you should be able to repair the automobile. Some say you must sacrifice the Highwind, too, but then you couldn't get to the Crater.

Revive Aeris: This rumor says that you can get some type of anti-death materia and get past the fish that guards the bottom of the City of Ancients. Another version says that Squaresoft was working on this, but never got it finished. A slightly different rumor says that if you do something to the ghost of Aeris, in her church (which I have seen), you will be able to get her back, but only in battle.
Note: The ghost of Aeris only seems to appear the first time you go into the church.

The Goblin: Supposedly you can get a certain goblin in your party if you fight it on the Goblin Island. It supposedly has 9999 HP.

Note: If anything, I believe that you can probably get Sephiroth in your party once Aeris dies using the Game Shark only. This would fill up the extra character slot left by Aeris when you make your party. (There are nine in total: one for Nanaki, Cloud, Tifa, Cid, Barret, Cait Sith, Yuffie, Vincent, and Aeris.)

For Everyone Seeing This: If you know anything about these rumors, especially if you have a picture, please send it! DEFINITELY tell me if you know what the 1/35 soldiers, the Super Sweeper, or the Autograph do! I'm certain that they must do something, because otherwise, what's the point of getting/winning them...