Found the problem... the tutorial had us put the following in the code:
if (queryType != CompletionProvider.COMPLETION_QUERY_TYPE)
return null;
which forced it to return nothing after the first request. Commenting that out takes care of it, and lets the drop-down work as desired. Still trying to figure out why exactly that code was in the tutorial... He has an explanation for it, but I don't see how it's very useful most of the time.