You can win items with the Jump Rope mini game. I searched around and this is what I found:
20 times = 10 Gil
50 times = Cactuar Card
100 times = Genji Card
200 times = Alexandria Card
300 times = Tiger Racket Card
1000 times = Item King of Jump Ropes
As for the cards. I havent heard of anything happening if you get all of the cards. I personally don't bother that much with the game