Using a custom app, I overrode (is that a word?) the appearance/home.html template to point at my own url on the Advanced button ({% url 'my_overrides:search_advanced' search_model='documents.Document' %}). From there it was creating custom versions of AdvancedSearchForm, AdvancedSearchView, and ResultsView to get what I wanted. My version of AdvancedSearchForm adds a ModelMultipleChoiceField which provides an autocomplete way to find specific documents.
In my AdvancedSearchForm, I was able to change the label on a field - Label - to Name (users were confused - "where's the document name?")