It seems that when one disables a module which has a customized menu item (for example, I moved the menu item for Advanced Profile settings to the Users admin menu), a situation occurs where menu corruption starts to take hold. First, the custom menu item remains for the disabled module. Then when I clicked "Reset" on the item wanting to remove it, the result was a duplicate item of "Site configuration" was created.
I then tried re-enabling the module, and the module's menu item was recreated but at the root of /admin, rather than where it was defined in admin/settings. There is no "reset" option here, either. And the bogus duplicate of "Site configuration" is still there, also without any Reset or Delete functionality.
Flushing the cache and executing devel's "rebuild menus" seems to have no effect.