Gold Saucer Battle Square Guide & Rewards
The Battle Square is a series of 8 battles that you can participate in on a single character. Each battle applies a different handicap to your character making them weaker as time goes on. The more battles you survive and the more damaging the handicaps that are used on you, the more BP you will receive.
BP is much like GP, it can be redeemed for a variety of different rewards that change 3 times throughout the game. The rewards shown in the screenshots above are the ones available after obtaining the Highwind on Disc 2. You must spend your BP before leaving the Battle Square or you will lose all of it.
However, you can participate in as many battles as you want and accumulate as much BP as you want, so long as you don't leave the area. What this means is technically you don't even have to be good at the Battle Square, so long as you participate in enough battles you'll be able to claim all of the rewards.
Battle Square Rewards (After obtaining Highwind):
Remedy: Cures all abnormal status effects
Enemy Lure Materia: Increases the chances of encountering enemies
Right Arm: Deals physical damage to all enemies in battle (about 1.6k)
Pre-Emptive Attack Materia: Increases your chances of getting a first strike in battle
Reagan Greens: Chocobos won't run from battle for 8 turns and they have an 87.5% of not running away after they finish eating
Speed Plus Materia: Increases a characters Dexterity
Stardust: Uses Comet2 on all opponents
Champion Belt: Accessory that increases Power and Vitality by 30
Omnislash: Cloud's Level 4 Limit Break
W-Summon Materia: Allows you to use 2 summons back to back during battle
I'd recommend getting a copy of every Materia available from the Battle Square as well as Omnislash, Cloud's Level 4 Limit Break. The W-Summon Materia allows you to use two summons in a row and is extremely useful for the harder bosses in the game such as Ruby Weapon and Emerald Weapon.
After obtaining the Omnislash and teaching it to Cloud as well as the W-Summon Materia the Battle Square will offer you a "Special Battle". This resembles the original Battle Square fights in every way except the enemies are much harder and the end boss is Proud Clod, the guy we fought in Midgar during Disc 2.
For the Special Battles every round is an enemy that you would have fought during the last round in the original Battle Square. As aforementioned the final boss is Proud Clod who probably won't be a challenge for you anymore either, depending on how well equipped you are.
If you need any advice for beating the Battle Square I recommend reading my Omnislash Level 4 Limit Break Guide. That gives a run down of the best Materia to use as well as strategies that you can use to beat this place very easily.