Location: Nanohana
Petitioner: Pompous Tam
Reward: Noble's Ring, 3,000 Berries
Objective Reward: Shabby Ball
In order to unlock this Side Story you need to complete the main story events of Alabasta and return to Waford. Shortly after doing so you'll be forced back into Alabasta, which will unlock a large number of new events throughout the kingdom. This includes More Important than Berries, and you can begin the Side Story by looking along the street to the east of the Spice Bean, in the northeast of town. The petitioner is standing on the side of the road.
The subject of this story is Pompous Tam, a jerkish noble who has lost a photograph while in the desert. He needs you to get it back. The photograph is in two pieces, so you'll need to hunt down both sides of the photograph to satisfy Tam. He suggests speaking to a caravan beyond the cave to the west of Nanohana. (And is none too polite if you have to ask him twice where to go next.)
Head west of Nanohana and into the Desolate Valley. The caravan is by the entrance in the south, and one of its members will tell you that the photograph flew into a nearby ravine. Head north from the caravan until you hit a campsite, then go northwest. You'll find a hole only Chopper can pass through. Beyond you'll find a huge skeleton, and on the other side of it is a broken bridge. In front of the bridge is a Picture of 2 Children.
Take it back to Tam and he'll verify that this is, indeed, the photo he wanted, and he's impatient to find the other piece. Speak to the caravan member again and he'll suggest you check the Ravine of No Return. Head there and you'll find a second caravan at the entrance. The guy in charge will say that he noticed a photograph in a spot with 'roaming scorpions and bones scattered around'.
Head east twice through the ravine until you find a clearing with two palm trees and a statue in the middle. South of the statue you'll find a collapsed man who will point you further south. (He's easy to miss if you focus on the Navi Bird too much, so be sure to look around.) You'll find a small clearing with two Lesser Brothers, many piles of bones, and a Picture of 1 Adult. Take this back to Tam to complete the Side Story and earn your reward.