Adaption to Chocobo Breeder

Another possible thing to add to chocobos would be a 'social' training ground. In this area your chocobo would have to navigate mazes or such and find moogles within the maze that would give it hints as to how to get out. This would increase the chocobos intellect and social stats.
Chocobos high in these stats would have a special use. You would be able to walk around cities/areas with it and use it as a sort of 'bargaining friend' allowing you to haggle with vendors using the 'synthesis window' in an attempt to buy something at a cheaper price, or perhaps use it on any NPC of your choice which could result in the NPC giving you items or quests. When using it on a normal NPC your 'synthesis window' would pop up with the options "Funny" "Cute" and "Intimidate", when you make a choice your chocobo would do an action to go with the option which would result in increasing your progress bar by a percentage, depending on success. At 100% the NPC would offer something to you, but if your Social percentage drops to 0 you fail the negotiation and cannot haggle with that NPC for another 12 hours (To go with the leve's reset timer after 1.19). you could also aquire a High Quality result if your 'Haggle' counter is high enough, resulting in better items or a better quest.

Incubating eggs could also operate on a similar method, using options such as "Warm", "Hot" and "Very hot" to increase your progress bar. Failure would result in the egg going to waste, where as if your Incubation Quality is high enough, could result in a High Quality Chocobo with better stats or abilities.

Leve

A Chocobo Breeder leve would be something pretty simple like hatching leve aquired eggs at your stables. Rewards would range from food and eggs to possibly tools and equipment.

A Chocobo Hunter leve would be to go out and capture chocobos, much like how mining leves are to go and mine mining spots, botany leves have you logging trees etc.