Here's perhaps a long shot but when I had a similar problem, it was a bad contact on the starter relay. The low current connection on the relay (to the solenoid - NOT the battery) actually gave-up.
It was a spent rivet if I recall correctly, that would make and break contact intermittently with vibrations.
On their death bed, nobody ever said, "Gee I wish I'd spent more time at work!"
'68 Sprite 1275
'76 Midget 1500