It's said that the Eternal Crystal - automatically in your inventory if you have a FF XIII save - will improve your chances on getting the Lucky Coin Fragment.... not by much, apparently, but a little.
My tip is either buy tons of Chips (or get them via Sazh's DLC Episode) and then put a rubber band over LB when the machine is On Fire and leave it like that, go do some other stuff. Hopefully, when you come back, you'll have earned it.
From what I remember, you can't miss getting the Lucky Coin Fragment cause there's a short dialogue appearing when you get it and you have to press A to continue...
Cloud and Sephiroth - battling each other since '97 for your amusement Give it a rest already!
Or do you just want to have coins? In that case, the racings are your way to go. Grab a silver chocobo, it's hidden in Academa 4XX AF. It's inside the sphere, were Hope and Alyssa are, chapter 4. Use Mog throw. Feed it with potent material only, so it will have attack and magic at 400 points at least.
Then go do some races. Your chocobo needs to be in the top 5 in order to give you some coins. And don't bet on yourself, you'll get a fair share of coins. Though it does take a while. It's also a good way to receive coins to waste at the slots, if you need the lucky coin. In case you don't want to buy the Sazh DLC.