I think I can help you on this.
My wife's car (Marea Weekend 2.00 HLX) experienced the same problem for about 2 years, last year the Fiat Garage 'reset' the lights (handbrake and ABS) and it got through the MOT, only for the lights to appear very soon after. This time it was not going to play ball, so I had to fix it.
First what was happening.
Both Handbrake and ABS light would be on for several days even at startup, then they would go out for maybe a day and then come on for several more, no apparent reason why. No bumps, sometimes would go out/ or on without even using the brakes.
What I learnt is that the ECU for the ABS checks on startup, so assuming the system is OK, then on startup the lights should appear for about 5 seconds (to show they are working) and then dissapear. So really, if the ABS is functioning normally then FIAT do not need to reset the lights. I also learnt that sometimes if you disconnected the battery then the lights would not appear immediately when reconnecting and turning on the ignition. But maybe on second start without even driving the lights would reappear. Total nightmare !
So, what did I do to sort this out.
First, as the system checks itself on startup, first thing I did was check the wheel sensors. The connections can be found as a yellow barrel in the engine compartment for the fronts (back left and back right, this one being near fuses), the rear sensors connections are located under rear seats. With ignition off, use a multimeter set to diode mode and check the resistance. If there is no reading swap the probes over. I got approx 750 for each of my sensors. Working on the principle that the lights were on at startup and the system checks itself when stationary and the sensors were giving back a signal, the I reckon they were at least working. So other then checking the cable for breaks (which was a waste of time), i left these alone.
Next I checked the brake fluid sensor, it is very basic and all seemed ok. It is basically opencircuit when the fluid level is good, so I guess even with the sensor disconnected the system will still think it is OK (although I have not checked this).
Working on the principle that the lights would sometimes be on at startup and sometimes off, 99% time being on, then i guessed the rest of the wiring was OK (or hoped as this would be a nightmare to fix!)
What was left. The infamous ECU. The one on my wife's car seems to be made by a company called Kelly-Hayes, you can see the part of the name written on the black part of the ECU if you look at it from the top. Changing the whole ABS pump seemed a daunting task as I don't fancy bleeding the system, so working on the fact it was some kind of electrical intermittant problem, after all the lights would go on/off without even braking, I decided to see if I could change just the electrical bit.
Procedure was roughly as follows;
1. Disconnect earth on battery
2. Undo the two bolts holding the brake fluid reseviour, purely to give me a little more room.
3. Undid the fuse lid and also the fuse bracket below.
4. removed the top part of the air filter box, which enabled me to move the big air pip which was getting in the way.
5. disconnect the ECU wiring loom which has a bosch marking on it. There is a plastic slide on the left hand side which you have to pull with a screwdriver further left and it will automatically remove the harness for you.
6. Once you have access, then there are 4 x torx screws which hold the top electronics to the valve assembly, remove all 4 of these.
7. Now you should have enough access to remove vertically the electronics, there is a rubber seal which should come out with it. There will be several wiring bits in the way, so be carefull.
8. steps 2-4 were purely there to give me more room. my wife's car is a 2.0 and the engine bay is really tight.
Next was to find a scrap marea with ABS. I was lucky enough to find one in my local garage without a front end smash which had a complete ABS unit including pump. I removed the lot using the same method, except where I couldn't undo the brake hoses on the valve unit, i cut them with pliers. Cost of unit £45
Then, i transferred the electronics bit from the scrap one and put it onto the pump, being careful of the seal. rest was reverse of above, putting everything back together.
Switched on car, ABS and hanbrake light went on for 5 seconds, then off and not come on again whilst driving.
Unit was a Kelly-Hayes part number KH13091804 which was the same as the one i replaced it with.
Hope this helps somewhat, even though my wife says i have written a load of windbag until I got to the end which is all you need to know. Sorry!
The end.