Thanks for the offer - I just have to figure out how to escape from the kids long enough
I landed myself in that situation so many times with my first X1/9 (along with other problems) and at the time never quite figured it out.
Reconditioning the starter motor is definitely a good idea if only to ensure that it is working so you can exclude it from the list of possibilities should it happen again.
One of the most substantial issues with the car (in general) is that despite the Italians flair for electronics they never quite got to grips with the longevity issue at that time (and still have problems today). The complexity of the wiring compared with contemporary cars from other manufacturers is astounding which of course makes it even more complex to track down faults but from personal experience it normally comes down to just a few things.
1) Poor earths - this has more to do with corrosion than poor wiring but without a decent earthing point every circuit in the car will perform poorly
2) Water ingress - condensation in the cabin results in water creeping into the wiring and ultimately causes more corrosion. The biggest victim is the main power feed to the fuse box - replace this and usually everything springs back to life. To make things worse a leaky windscreen or heater matrix can spell disaster
3) Skimping on relays - despite the huge number of relays in the fusebox they were kept to the bare minimum. For most circuits it isn't an issue but for the headlights it is a disaster. Mid-life cars were given a pair of high-beam relays to improve visibility (remember that they used sealed tungsten lamps) while the later cars had relays for the dipped circuit as well. The reason for the upgrades is that the steering column switch tended to melt internally around the contacts. The telltale is watching the charge light when you switch the lights on. If the light comes on stronger with dipped than it does with full beam then you know that you're missing the dipped relays. This of course assumes that the charge light comes on when the lights are used (which sums up pretty much every X1/9 I've ever sat in). If the charge light doesn't appear then you probably do not need to upgrade any of the wiring as someone has done it for you already!
4) Bodged repairs - most auto-electricians just run away screaming when they see an X1/9 so it is normally down to the less qualified (or more likely unqualified) electricians to try and fix things when it goes wrong. The end result is hideous to put straight but it can be done with patience. The trick is to make sure you have the right version of the wiring loom diagrams. They changed regularly and on top of that there tends to be a degree of cheating at the factory when a particular colour wire ran out so tracing the wire you want can be entertaining to say the least.
5) Poor QA at the factory - modern wiring loom testing is automated and doesn't just test continuity, there are checks for impedence and interference that ensure the loom is properly assembled. A 70s or 80s Fiat doesn't have the benefit of such things and on top of that the connectors are little more than plastic shrouded crimped spade terminals, nothing like modern self-sealing connectors. Even the printed circuit boards used in the dashboard have been known to corrode so you can assume nothing about the quality of the wiring even if it is untouched.