It does sound like starter motor solenoid, although the radio things a bit wierd. But, as WB suggested, the starter motor, and battery are pretty obvious suspects. A new gear reduction starter motor is pretty cheap these days, about 130 last I looked but that was a while back, (they're nice and short, its a good peice of gear to have.)

Times like that its often best to check everything over from the beginning, is the negative lead to the block connected propery, is it a good clean conection, whats happening at the back of the alernator, is everything done up right and in the right place, etc. What are the starter connections like etc. A nice chunky modern earth strap from the block to the chassis can sometimes also help sort weird ghost problems in older cars, you install it to an already exsisting fastner on the motor and the chassis/body at each end.