Here is a list of rumors about hidden secrets that may be able to be accomplished/obtained in Final Fantasy 7. Most or all of these are not true, they are just rumors.
Bring Aeris back to life without a gameshark
I have heard several methods of how to revive Aeris. The most common idea is to help the ill man in the pipe in the slums near Aeris's house. I did this before I heard the rumor, and it didn't seem to have an effect. There are many things that can be done in the Wall Market however, and there may be a slim chance that Aeris can be brought back to life here.
Cloud's 5th Limit Break
Some web sites claim that Cloud has a fifth limit break, that does 99999 (5 digits) damage with each attack.
Play as Sephiroth
There is a rumor stating that you could get Sephiroth in your party somehow, using the Masumune (item) that can be won in the Gold Saucer.
White Chocobo and Onyx Weapon
There is a rumor that by breeding S-Class chocobos that have a speed of over 175, you can breed a white chocobo that can go underwater. It can also go 'under' the marsh where the Midgar Zolom is, and there is an 'Onyx Weapon' there, more powerful than Ruby or Emerald Weapons.