I don't know how to add this to the Quest List so if anyone comes across this please edit it.
Valderotaux is the person with the Lost Chocobo as mentioned by the Chocobo caretaker Hantileon in San d'Oria.
After Hantileon tells you that all he knows about the lost chocobo owner is a Elder Elvvan Merchant with the first letter of his name starting with a V; come talk to Valderotaux.
Valderotaux will then proceed to give you Key Item: Story of a Diligent Chocobo
I don't know what the story does for the chocobo's attribute, but if I choose to tell this story, I will report with my findings.
Hantileon might assign a different NPC to the lost chocobo, but this is what I got.