To unlock these weapons you must return to dungeons you've already beat and defeat the extra boss that appears.
Kanji's Secret Weapon - Defeat the extra boss in Steamy Bath House
Kuma's Secret Weapon - Defeat the extra boss in Opera House
Main Character's Secret Weapon - Defeat the extra boss in Void Quest
Naoto's Secret Weapon - Defeat the extra boss in Secret Lab
Yousuke's Secret Weapon - Found in the boss's den in Mysterious Shopping Street
Yukiko's Secret Weapon - Defeat the extra boss in Yukiko's Castle