Lexus GX 2004-09 :: GX470 Won't Start - No Solenoid Click / No Starter Noise

Our 2004 GX has been a great car, but is suddenly doing a number of strange things - like not start. When I say it won't start, I mean you can turn the key, the dash lights up, the steering wheel adjusts, radio goes on, etc, but then NOTHING! No solenoid "click", no starter noise, NOTHING! Does not crank at all. The car just turned 60,000 miles, is in great shape, lives in the garage and has never been in an accident.

After the first couple of times it did this, had the battery replaced, as it was getting close. and then the car started again. Until the next day. No rhyme or reason as to when it would do this. It might start 10 times in a row, then NOT start for a while, then magically fire up instantly, even after several key turns. I wiggled the key, tried a different key, shifted to Neutral, wiggled the shifter to try to rule out security interlock, bad ignition switch and shift interlock. No dice.

Did some more research and thought it might be the starter relay - banged on it and it started, so replaced that part. Didn't solve the problem. It also began to randomly die while idling (in park, or in drive). Sometimes it would re-start, sometimes not. The times it has done this, it typically dies after running or 15-20 minutes, but NOT ALWAYS.... Sometimes it will die once, then be good for the rest of the day. Sometimes it re-starts immediately, after a few minutes, or more recently only after sitting for a few hours.. So basically we can't use the car.