Rather than "difficult content" I'd like to see solo dungeons you can do with your chocobo out, also unsynched dungeons with only your chocobo would become "solo dungeons".
Would also be nice if this was an opportunity to obtain various chocobo related stuff like bardings or various food.

I don't think we need more difficult content, because we do have that in game currently. What we're lacking is stuff to do with the companion. Right now, it's more of a nuisance than a companion IMHO, as you have to store it to ride another mount or queue up for the dungeon finder. I sometimes find myself burning through 20 grysal greens a session because of that.
What's the deal with those grysals anyways? You throw cash out of the window to have your choco with you for a few minutes in most cases.