When you select a body armor, Brygid demands the corresponding subligar. These are dropped by Fomors in Phomiuna Aqueducts as well as in the Sacrarium. All of these subligars are level 50 and Rare/EX, so you must personally go acquire it.
Once you request a specific body armor, you are locked into that request until you trade her the subligar for it.
If you leave Bastok Markets after showing Brygid the armor she requests but before choosing the body armor you want, you must wear the requested armor again when you return before you can choose your armor.
Trade the requested subligar to Brygid receive your new body armor.
This quest's name is an allusion to Star Wars Episode VI: Return of the Jedi, while Brygid the Stylist Strikes Back is an allusion to Star Wars Episode V: The Empire Strikes Back.