OK, so here's what I found last night. The click is coming from a box, about 2" by 2.5", hanging right above the battery on the left side of the bike. Four wires lead into the top with a thick plastic clip around them. You have to pry up one of those recalcitrant plastic latches to detach it. The inside is burnt to shit and there is corrosion everywhere (fault of the side cover missing). Me smell homebrew electrical repair from the previous owner. Fortunately, I have a soldering iron and nothing looks too complicated. I can update with pics when I get home.