Well, there are a lot of ideas and advice going around for something mysterious, as you describe. Here goes one more, based on personal experience of the exact same problem, but in a Uno.
I don't know the mechanics of the Stilo but if it is anything like the Uno, where the Electronic Ignition Module is mounted on the distributor, then it is likely to suffer from the same problem, electronics overheating. Over time a silicon component fault may develop and the Ignition Module will behave somewhat erratically. The matter can be checked by connecting an oscilloscope to the coil positive, at the exact time when the engine refuses to start. The ignition pulse trace will appear irregular and untimely, all over the place... Another way of testing the theory is to get a new Module and hopefully, the problem will go away. By the way, if you could mount the Ignition Module away from the engine, somewhere cool, the problem is unlikely to happen again.
Best of luck or just be the best!