Intermittent starting problem - 94 Dodge Intrepid (3.3 engine)

Lately, I have been experiencing an intermittent starting problem. Every once in a while, it won't start, even though the starter turns the engine over.
The battery is fine. There are no codes in the computer, and the fuel pump has just been changed and the electrical system checked out, but with no warning at all it fails to start about every fifth use.

I find sometimes that if I remove the key, reinsert it, and try again, it will start right away. Other times, after 4 or 5 unsuccessful tries, I just leave it and after I wait 20 minutes or so, it starts, no problem. It has been kept inside in a heated garage to dry it out thoroughly, but to no avail. There is just no predicting when it will or won't start.

On a normal start, all the warning lights on the dashboard light up, but on those occasions when it does not start, only the oil pressure warning light comes on.

Any help would be appreciated.

There might be an issue with your ignition switch? You need to do some testing, when it refuses to start. Turn the key on, see if you have any reference voltage at one of the engine sensors. It doesn't matter which one, whatever is easy to get at. Crank sensor--cam sensor---throttle position sensor. Unplug the sensor, turn on the key, check for voltage at one of the terminals. If there is no voltage, check for feed going to the pcm. My info showed two feeds, one is fuse B 20amp--in the power distribution center under the hood, hot all the time. The other is fuse cavity 12 10amp in the fuse box, goes hot when you turn the key on. You have to do the testing, when it refuses to start, you can pull those fuses, use a testlite to see if one of the terminals for each fuse is hot. One fuse is hot all the time, the other goes hot when you turn the on.