After a bunch of diagnostics, I narrowed the problem down to a high resistance connection in the convertible top "raise" circuitry. At some point in the car's past, a wiring repair was made to the "top up" relay wiring, which can be seen in the above photo, (note the blue crimp connectors toward the top center). This repair was made long enough ago that corrosion had developed inside one of the crimp connectors. I made the proper repair, (soldered connections and heat shrink insulation), which solved the problem.
The customer had a shop manual for the car, which was good because we didn't. The shop manual for the modern Ford product is generally about 3 inches thick, and has a separate 1 inch thick book for all the wiring diagrams. In 1964, wiring diagrams for everything on the car took up about 10-15 pages in the back of the 1 inch thick shop manual.