A Memorable Hat is a sidequest you can find in Star Ocean: The Divine Force and pick up in Seaport Of Cotto.
If you head towards the item stall, you can find Old Woman Preparing For A Trip. She has lost her hat and wants you to go and retrieve it for her.
You can find the A Lady's Hat item by heading to Cottorinth Summit Approach. You'll find the item as a green glowing item drop at the center of where you once fought the Holy Bird boss.
Once found, return to Old Woman Preparing For A Trip and complete the quest for your rewards.
Reward: