Very very common problem on the Stilo, I've had two and both have had this happen.
Its a reverse switch, so with the switch pressed in the lights are off and vice versa, this is why you are finding the lights do actually go off when you pull the brake lever up.
What's happening is the switch is wearing out and the spring inside it is getting weak.
Anyway, never have I replaced the switch in either of my two Stilos. There are three ways to fix this.
1. The easiest way is to get a coin or two and stick them to the small plate on the brake lever which presses the switch in with a little bit of black tape this will pack it out enough to stop the problem.
2. Bend the small plate on the brake lever towards the switch a little more, this is a careful process however as you don't want to move it too far and you also don't want to break it off as it is welded on.
3. Replace the switch totally. Fiat would recommend the right one. Personally I just do a bit of DIY.
Hope one of these methods works for you.