The noise could be a relay that is getting pulled in, then dropping out again – which would cause the clicking sound. Most cars have a load reduction relay, and there may be others as well, that apply power only when the ignition is on.
Usually the clicking sound happens when the battery is very low, so it is a bit puzzling that you're hearing this when the battery seems to get good.
Some cars also have valves in their vacuum systems that modulate the parameter they are controlling (turbo boost, EGR, etc.) by opening and closing relatively frequently (several times per second). In my experience that sound can be heard, but requires careful listening.
Unless this sound is typical of Fiats and somebody recognizes it, the next thing to do would be for you to try and isolate the sound. Since it happens with the engine off, it should be fairly easy to do this, and relatively free of danger. Here's what to do:
- Open the hood and front doors.
- Turn the ignition on and listen for the sound.
- Move around listening to the sound so that you can identify the general area it comes from. The area around the steering wheel, the fuse panel, and the firewall are particularly worth paying attention to.
- Once you localize the source, start looking for a little box with wires coming from it. It will look something like the ones to above and to the right of the yellow arrows in the photo below:
Also listen around the brake booster (the large round thing more or less directly in front of the driver) for noise coming from the ABS controller and the belts on the engine for noise coming from the A/C compressor – really just keep listening until you're pretty sure you know where the sound is coming from.
If you can isolate it well enough that you think you see the source that is great, but even if you can't it will help tremendously to know about where it comes from (e.g., above the pedals, in the area around the fuse panel).
If you can point to what you think is the source. You should be able to confirm this by holding a "dense rod" against it and listening to the other end. As long screwdriver works well for this, but anything dense will conduct sound pretty well. If the "stethoscope" is touching the source you will hear a very distinct click.
Send us a photo of the culprit.