I know some....

Carbuncle + Diamond, Moonstone, Emerald = protect, shell, reflect and vanish can be cast base on the stone you used

Fenrir + Maiden Prayer = Cause Wind Damage, Millenial Decay.

Odin + Ancient Aroma = Gives Odin's Sword ability to Dagger which increases the chance of killing the enemies, about giving the wind damage, I don't know since I'm not putting Dagger in my party (I put Eiko). I only read the description.

Pumice Piece + Summoners = They call forth an Eidolons with full scene

Phoenix + Phoenix Pinion = every phoenix pinion will add 2% of getting auto-life phoenix (just guessing, I heard it somewhere)

Leviathan, Ifrit, Ramuh, Shiva, Bahamut, Atomos, Madeen and Ark damage maybe depends on their stone.

Leviathan power + Aquamarine = dmg.
Ifrit power + Topaz = dmg.
Ramuh power + Peridot = dmg.
Shiva power + Opal = dmg.
Bahamut power + Garnet = dmg.
Atomos power + Amethyst = dmg.
Madeen power + Ribbon = dmg.
Ark power + Lapis Lazuli = dmg.

I'm just guessing, don't take it seriously.