Treasured Ball is probably the shortest and the most accessible Lightning Returns: Final Fantasy XIII side-quest available in the area known as Luxerion.
Nonetheless, Treasured Ball must be completed if you want to increase the amount of time Lightning has until the world ends; therefore, you should address it as soon as you finish Hunter In Light And Shadow, by visiting the South Station.
Even if it is an accessible side-quest, if you encounter any difficulties in completing it, you can use the following Lightning Returns: Final Fantasy 13 guide, which was created specifically for the Treasure Ball side-quest.
Description: “On 1st Avenue near Luxerion’s South Station, Lightning encounters a boy who has lost his ball on a rooftop. It looks like Lightning can reach the roof from the top of a nearby stone wall. Maybe she could fetch the rubber ball for the boy.”
Requirements: Complete Main Quest 1-5: Hunter In Light And Shadow.
Start the quest at 10 AM and finish it before 8 AM.
Rewards: 100 Gil, Maximum HP + 30, Strength + 3, Salvage Pilot’s Badge
Treasured Ball doesn’t require your presence in other areas, that’s why it is a very short quest, but you will have to discover it, by finding the client, or the quest-giver.
Make sure you meet all requirements above, and then visit the South Station. Here you will have to find Talbot.
While near the Canvas of Prayers in the South Station area, move north and you will enter 1st Ave. Look for a boy with brown hair and black jacket, near a structure made of wood and a pole. Talk to him and you will be asked to fetch his Rubber Ball.
The Rubber Ball is located on top of the structure, but you won’t be able to pick it up while being below its location.
To reach it, turn around and follow the stairs nearby. Turn right to get to the upper wall, and then jump on the platform.
The Rubber Ball glows on the ground. Take it and then return to the boy, to complete the quest and receive your rewards.