[DeTomaso] intermittent non-starting car NPC

[pantdino at aol.com]

I'm hoping you guys can give some more ideas.
My son's 1995 Lexus ES300 (fancy Camry 6 cyl, 219K miles) refuses to start sometimes.  About a year ago it was obviously a battery power problem, as the headlights would come on very dimly.  Turned out to be a corroded negative ground cable and replacing it solved the problem (also put in new battery).

But lately it sounds to me like the ignition switch or starter relay. My son reports that --occasionally-- turning the key to start mode causes NOTHING to happen-- no clicking, no turning over, nothing.
I have not been present at these times.  Once I arrived some hours after the car refused to start and the headlights were bright and turning the key caused the engine to turn over very briskly and start immediately.  I have tried wriggling the ignition key and the battery terminals to see if I could see the circuit cut out and they didn't 

My son usually gets a jump, which works, but I don't think the battery is the problem so I think it's just chance that the circuit is completed that time.

Anyone know if 1995 Japanese cars had starter solenoids (I mean separate, like the old Mustangs), and if they can fail intermittently?  

As I write this I realize I need to quiz him if the failures only occur when the car is hot, as I have heard of electrical components that fail when hot and are OK again once they cool off. So by the time the friend arrives to give him a jump or I get home it has cooled off.   I'm thinking maybe just replace the starter motor / solenoid unit as the most likely cause?

I have a workshop manual on CD for the car but it's crappy.

Thanks

Jim