On the snow mobile when on the snow plains, hold x and tap circle until a new screen shows up. This screen is a mini-game: get as many matches as you can to win prizes!
To unlock extras in the game (an art gallery, a bestiary, and a music player), simply beat the game.
In Final Fantasy II on the PSP, you can unlock Soul of Rebirth if you have beaten the game once. Soul of Rebirth allows you to play a separate scenario with Minwu, Josef, Scott, and Ricard. This is made to be a challenge, as the time put into this scenario will mainly be from making your characters stronger.