I would have liked some sorta Gold Saucer place like in FFVII, but with even more mini games. Mind you I guess you would call that more of a mini game than a sidequest.

The best sidequest imo in X was definately the Monster Arena, if you can call that a sidequest (I would) it was pretty challenging and time consuming but I also found it fun and it had some very good rewards.

As for an actual sidequest I'm not sure, maybe something to do with chocobos or mogs would be fun. How about breeding mogs we haven't had that yet, lol.