I took my son’s car in yesterday. Of course, they had no trouble starting it at the shop (isn’t that always the way?) so they didn’t get to experience it first-hand. However, they checked everything out and said they thought it was the battery, the battery was “borderline”. Everything else tested ok. So today he’s got a new battery (plus new front brakes), hopefully that solves the problem.
Thanks to all who responded!
Probably one of these?
Don’t worry - the European versions were considered crap as well. 
Ironically, the starter motor was very likely a Bosch one - Geman built.
I’ve had an old Peugeot in the past that sometimes wouldn’t crank over, and indeed a swift hit with a hammer did the trick every time. Dallas Jones analysed it correctly a few posts above.
DivideComedienne, good to hear the car seems to behave again. Fingers crossed.
modern batteries don’t fade away like their predecessors did 30 years ago. They are smaller and much more powerful. This is accomplished by adding more plates and spacing them closer. When they go they go quickly so if they say it’s borderline then it’s getting ready to self destruct. I’ve had many a battery die in mid-start with no warning.
On three of my cars with Japanese starters, I’ve had a mysterious intermittent problem where you’d turn the key and nothing happens. Nothing, no cranking, no nothing. Acts like a dead battery, but the battery is new and fully charged.
The problem eventually turned out to be the starter motor solenoid. A couple contacts inside get eaten away, and every now and then they don’t make contact like they should. The parts are about $4 at an auto electric shop, if you feel confident about tearing apart the starter; or you could get a rebuilt starter.