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To unlock this, obtain every piece of head gear at some point in the game. You do not need to keep them after acquiring them. There are four pieces that are drops and will require some grinding to get.

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Bandanna - Potos
Hair Ribbon - Dwarf Village
Rabbite Cap - Pandora
Head Gear - Dwarf Village
Quill Cap - Neko (Moogle Village)
Steel Cap - Mantago
Golden Tiara - Kakkara
Racoon Cap - Kakkara
Quilted Hood - Northtown
Tiger Cap - Northtown
Circlet - Mandala
Unicorn Helm - Gold City
Dragon Helm - Gold City
Duck Helm - Gold City
Needle Helm - Neko (Mana Palace)

Dropped
Cockatrice Cap - Basilisk (Underground City)
Amulet Helm - National Scar (Mana Palace)
*Griffin Helm - Griffin Claw (Pure Land)
Faerie Crown - Mace Menace (Mana Fortress)

*Griffin Helm is missable. Get this before you finish the Pure Lands!