When you first visit Junon, there are 2 "mini-games" which can earn you different rewards, depending on how well you perform.
After disguising as a Shinra MP, you can take part in the Shinra March. How well you do affects the television ratings. Ratings of 0-29% earn you a Grenade. 30-39% earn you 6 Potions. 40-49% earns you 6 Ethers, while 50% or higher earns you 5000 Gil. Personally I find this very difficult.
Subsequently, during President Rufus' send-off, you can earn the following: 0-50 points earns you a Silver Glasses accessory, 60-90 points earns you a HP Plus materia, and 100 points or more earns you the Force Stealer sword. To earn high points, press the correct buttons to do the correct poses as quickly as possible once the button icons display on screen. You get extra points for doing a pose faster than your fellow soldiers. Personally I find this very easy.
N.B. It is debatable that a HP Plus materia is actually more useful than a Force Stealer sword, even though you need less points for the materia.
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