I almost fell into the trap of naming it Boco.
Thankfully I didn't because it seems like a billion people have played Final Fantasy Tactics.
Before that game came out, Chocobos were merely thought of as nameless beasts too unimportant to personalize.

I ended up naming mine just the generic and plain "Chocobo". It's wayyy more unique than you think actually. If Ash can name his Pikachu "Pikachu", I don't see why I can't name my Chocobo "Chocobo".