What do you do in a Views field theming function to change the output?
I seem to have stumbled into something that works:

In the Views control panel for my Calendar view, I went down into Format > Settings > Customize Fields, and clicked the "Use a custom title" checkbox.  After I did this, the system started paying attention to views-view-field--fullcalendar--field-target-date.tpl.php, and used whatever I printed as the title of the events in the calendar.  

So far, all is well, although I'm not sure whether this is an inherent part of the Views system per se, or if it has something to do with the module's use of the fullcalendar jQuery plugin.  Other solutions/answers/explanations are of course welcome.