I would like to see

Breeding (But MUST be improved from 11 and be FUN this time and not cost an effing fortune)
Racing (like in 7, NOT 11)
Variety of colors and other physical differences
Personalities
Leveling chocobos
Chocobo knights
Chocobo packs for extra storage
Equipable armor and accessories for chocobos
Able to fight both ON and alongside the chocobo
Summonable and rideable anywhere except in cities
Have to feed them (not on a timely basis, but like repairing gear) or their stats lower.
Differently bred chocos have different stat builds and essentially become different 'classes' of chocos (healer, melee, etc)
Able to own more than one chocobo, but not an unlimited amount. Maybe 3-5.
No chocos slower than rentals
Completely free naming for chocos (NOT chosen from a list)
A 'companion' feel to them. Like they're not just an instrument to ride on, but a beloved pet. Walk it around alongside you and such.