Steve and Deb,
MDL-10993 "Restore creates duplicate activity/resource icons with no text" sounds related. Do you see two icons for every activity/resource, one with text and one without, or just one icon without text? if you think your problem is related, please comment on MDL-10993.
Deb's solution sounds like a workaround. Normally it should be possible to just check what you want; backup should be able to figure out the dependencies. For example, if you check quiz, backup will take questions from the database.
Backup and restore
Backup function won't save Topic titles or text
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