As we were about to begin another start attempt, this time with maintenance supervision in the cockpit, a woman approached me in the forward galley area. With a distraught expression and a plastic cup filled with an unknown liquid ingredient over ice, she told me that she felt unsafe and that a friend seated next to her was "freaking out." Although I had made numerous PA announcements explaining our predicament, it didn't seem to matter. The woman was not satisfied. She followed me into the cockpit without an invitation. When I offered her a more detailed explanation, she indicated a desire to deplane. I assured her that the problem did not render the airplane unairworthy. She could deplane if that was her option. Although she was astonished by the fact that we wouldn't incur an additional delay to locate her bag among the bags of 187 other people, she summoned the courage to remain on board. I smiled, slid into my seat, and politely excused myself to attend to less important matters … like starting an engine.