Wow, this is a good one.
I love thinking about problems like this one. .
when it is someone else's car. :)
I just wish I was familiar with your system.

The first thing that comes to mind when you say that it starts only after sitting all day is gasoline-fouled spark plugs. After it dies, examine a spark plug for wetness, and check that spark again to make sure you still have spark. Fouled spark plugs (assuming you have the correct plugs) would be a symptom rather than the root cause, but it would explain why the car does not start. If they are fouled and you still have spark, suspect the fuel pressure regulator.

After that, electrical resistance almost anywhere that causes a connection/wire/component to heat up and stop conducting. Most likely in the ignition, but if you still have spark during no-start, then it has to be in the fuel control system somewhere. The AMM is the place to start looking. The problem with that theory is that you'd expect the car to restart after cooling for only a few minutes.