This could be one of 2 things.
1. They key has actually failed, although the light might still be working. First have you tried replacing the battery?
2. The Car has "forgotten" the remote code, so pressing the button results in nothing. If you've got a model other than the 1.2 Active you should have 2 remote keys, does one work and the other one not work?
If like me you have the 1.2 Active with 1 remote key. The fix is as follows..... and its going to cost you!
1. Go to your local Fiat garage and request your vehicles security code, they can look it up for you, but it will cost you about £10. Its your code though, so when you get it right it down, and keep it safe.
2. You need a copy of FiatECUscan, which last time I checked costs about £40 for the full version (the free version lets you play with lots of things but not central locking codes).
3. an ODBII cable, go for a USB one, the bluetooth ones seem a good deal but rarely work properly.
Once you have all of these, use FiatECUscan to re-code your key to your car. Its really easy. I can't remember the exact steps but its on the FiatECUscan forum.
All in this will cost you about £60-70, but you'll have kit you can re-use come service time etc.
Or ask around here if anyone is local to you with FiatECUscan and an ODBII who could help, you'll still have to fork out for the cars security code. Best case you'll be able to fix it for £10.
Or, you could go to a Fiat Garage for the whole shebang, they will charge £150 for new Key (which you don't need), £10 for the code and probably an hours labour.
I urge you to consider doing it yourself, because FiatECUscan is pretty sweet, lets you reset service indicator after self servicing. Lets you check and test components of the car, like checking alternator output, lights, fans, relays all kinds.
And finally once you get your key working again you can set up your car to flash its indicators when you remote lock the car.