I figured out what was up. I had a behavior in the page that was refreshing the div containing the flaky dialog, but without refreshing the entire page. So when you first visit the page, the jQuery dialog function set up the dialog, shuffled around the DOM and all that, and then it works. When you hit one of the actions that refreshed that div, it must have broken the linkage between the jQuery dialog and the div where it had moved it within the page, so I'm guessing that the datePicker was attached to the dialog in the copy.
I changed the other behavior so that instead of refreshing the entire div, it only refreshes the part that contains a list of items, so it doesn't refresh the div that the jQuery dialog (and the associated jQuery datepicker) are attached to. Now it works like it should.
Hopefully this will help someone else somewhere down the road!
Thanks!