(I believe) After getting Omnislash, a special battle option will be available in the arena. After winning it, you can get new materia, and I'm pretty sure that's one of the prizes. (CIW.)
Okay, here it is, thanks to CoN of course.
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Also, if you get the Ultima Weapon, Omnislash, and the W-Summon, you can participate in the Special Battles by speaking with the receptionist. Only Cloud can participate to this battle. The battles are harder and the prizes better in this elite class (well, one of them is), if you can win. Here are the prizes and the enemies you'll face:
* Final Attack: After first victory
* Autograph: After second victory
* Combat Diary: After third victory
* Gambler: After fourth victory
After you win four times, you'll simply randomly receive either an Autograph, Combat Diary, or a Gambler, but sometimes you can win a Ribbon. It's even rarer than these other items, however, so be patient if you really want to win one this way.
* First battle: Sea Worm
* Second battle: Ho-Chu
* Third battle: Unknown 3
* Fourth battle: Serpent
* Fifth battle: Wolfmeister
* Sixth battle: Behemoth
* Seventh battle: Maximum Kimaira
* Final battle: Proud Clod
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The 'special' battle isn't really that much harder.