If you have either Tifa or Cid in your party when you go to race chocobos, they might ask to ride the chocobo. It doesn't really matter unless you're sick of looking at Cloud ride them all the time. If both are in the party, Tifa takes precedence.
They might also ask to ride the snowboard if you play that minigame over in Wonder Sqaure. Yeah, it's a lot of fun watching Cid take a drag on his cigarette right before barreling down the Crazy Course at top speed.