Tom-
Very first things to check are the battery terminals. They need to be clean and tight, no wiggle. The cables should be securely connected to the clamps, with no corrosion. Sometimes people cut the lead and install new cable clamps and they corrode after a while (BTDT).
Next, check the -ve ground lead to the chassis from the battery. Same deal, it should be clean and tight against the chassis. I always wrap my wrench in electrical tape when working in the confines of the B's battery box. One misstep on the +ve to ground and you'll weld the wrench to the car. Not fun.
Next, trace the battery +ve cable forward from the battery under the passenger floor pan to the starter solenoid lug. Check carefully for damage/cracks/whatever. The cable ends at the starter solenoid and should be, you guessed it ... clean and tight. Sometimes the copper stud on the solenoid is stripped or corroded or both. Disconnect the -ve battery terminal before you attempt to remove and clean it.
That ought to keep you busy for an hour or two. Report back what you found and we'll go from there.
FWIW, this summer I replaced the main battery cable on my B with a pre-made assy from British Wiring. It fit perfectly and only took an hour or so to install. Might want to look into that. The new cable was a bit thinner. Just for the heck of it, I cut off the the coroded ends of the old cable and measured the resistance against the new one. The new (slightly thinner) cable was *half* the resistance of the old one, which suggests corrosion inside the sheath.