Switched the starter, still not starting

[bigbadjohn666]

Ok I have checked the battery and its fine, my father inlaw says starting problem has been a problem for some time now.

What happens is the car will start from cold no problem, always does and never fails, drive the car for a while and stop, when you go to start the car nothing, you can hear a faint clicking from the dash, but not a clicking from the solenoid, all the dash lights dim right out as you turn the key although the battery is not flat, as a rule of thumb if you wait 10 to 15 mins the car will then start no problem and you can turn off and start up again and again, somethimes it won't start for over an hour or more I have read the various possible causes but really would like to be sure what is actually causing this problem as I now avoid driving my Supra as its unreliable and erratic.

The car has covered 60k from new and in its time has had a head gasget and been dealer serviced on a regular basis.

Do I do the relay conversion ?

Do I check The Earth's ?

I have turned off the alarm ?

Does the Supra have a built in security system ?