Location: Qianling Pier
Petitioner: Liu Miaonü
Reward: Fanyang Straw Hat, Martial Arts Thoughts, 5,000 coins
To trigger this side quest you just need to find it. The petitioner, Liu Miaonü, lives in the small settlement of Qianling Pier, to the south of Wutong Village. To reach the pier you'll need to head a little ways west of Wutong, cross a bridge, then head south and southeast. Qianling Pier is on the banks of a river that flows south.
Speak to Miaonü - she's right by the village's entrance - and you'll find her crying. Apparently the bandits in the nearby Cave of Tribulations have stolen some of her fabrics, and she doesn't know what to do. You have two options at this point, though ultimately it boils down to retrieving the cloth for Miaonü.
To complete this quest you need to retrieve ten pieces of Satin. Although you're directed to the Cave of Tribulations, which is a short walk to the south, you can get Satin from most Bandit-type enemies of a particular strength. If you find the enemies in the cave a bit too strong (likely if you came here early on) you should wait until you have a temporary party member along for the ride. Early on this will usually be Mowen, who can handle the enemies in the Cave of Tribulations.
Return to Liu Miaoü with ten pieces of Satin to complete the quest, and receive your reward.