Left mouse click for external mouse not working after upgrading to Lion
The solution was to disable the internal trackpad. The trackpad, one of the older style pads with the separate button along the bottom edge, had a malfunctioning button and that seemed to be at the heart of the problem.

As soon as I disabled the trackpad the mouse would word. And as soon as I re-enabled the trackpad the mouse would cease to work.

For details on how to disable the trackpad see this question and answer.