Others may have more to add beyond what I write here or may have much better advice on doing this activity, hopefully they will chime in. We have a number of members who work in the automotive industry and electronic industry so they may know of better ways to do the following.
As you look around the car, you will see ‘ground blooms’ which collect a set of nearby ground wires from components to the body. There should be a list/image of the positions in the Fiat manual for your year which shows where to find them, they haven’t changed much over the years.
The connection of the bloom to the body of the car, which carries the electrons back to the battery, can become corroded and degrade the connection making the element looking for a ground work intermittently. The wires connected to the bloom can also have corrosion over time so cleaning the bloom and the wire spade connector will ensure you have a good ground. While doing this you will find broken wires, frayed wires or wire connections that need to be replaced.
Deoxit or similar contact cleaner with a brass brush are good for cleaning terminals. Try to manipulate the wires as little as is reasonable to avoid cracking old insulation etc. When putting connections back together using silicone dielectric grease will protect the connection from corrosion. When reattaching the bloom I use conductive grease between the bloom and the body and between the bloom and the fastener:
https://www.amazon.com/MG-Chemicals-Carbon-Conductive-Grease/dp/B00SMRNSR0
The ground bloom could well be corroded, new ones are avialable from Fiat vendors but you can also refurbish them if they are not too bad using a rust removal treatment after cleaning grease or other crap off with Simple Green or similar with a soft brush. I use Evaoporust to remove corrosion, from steel parts, it does a good job and won’t destroy the part. I bought some which come in a pail with a tray you can put parts into and easily retrieve them.
The reality is you will find broken wires, bad connectors etc as you do this so you will want to have a supply of proper spade connectors (
http://www.cycleterminal.com/ or other real electrical suppliers like Digikey or similar) and a good tool for crimping them. Avoid the connectors which have the insulator as part of the connector which you find at most autoparts stores, these rarely provide a good connection particularly if you are not using the correct crimper. Avoid being tempted into buying a kit of cheap connectors on Amazon or the crap at Autozone etc.
Doing all of this will greatly improve the function of the electrical components, effectively making them work like when the car was new and ready for another 40+ years.
Once you do that you can start in on all the individual electrical power connections, working around the car with each light etc.
Image of Ground Bloom courtesy of Mid West Bayless: