Well, the rabbit hole went deeper than I thought. Upon poking around, I discovered the Remote Admin Console. Even there, although I could check the Favorites button and get it to light up, that setting was not saved. So I resorted to exporting the entire address book (change Management to Export in the upper-left corner on the Address Book screen). Then I did a Delete All. I also updated the CSV file to make the new contact a favorite (although I could have done this later).
Then I re-imported the address book, using the replace contacts instead of add contacts option. Now I can make any contacts favorites.
Problem solved, although it does leave me wondering what more there is to the internal address book than what is actually exported, since the export file looked and functioned correctly, and it seems it was only the Delete All option that cleared things to make it all usable.