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I've the exact same symptoms as your client, but in this case I can't find any incorrect file permissions. And after a bunch of searching, your comment here is the first related info I've found.

Yep, common scenario.  ;-)  Tip: Always make sure you COPY files and folders into the 12-hive templates location rather than MOVING them from another location.  For example, if you have the css file sitting on your desktop and move it into the layouts/.../styles directory, it'll keep the same file permissions it had on your desktop, which are incorrect.  If you COPY the file, however, the file permissions won't copy over, they'll automatically inherit the appropriate permissions.  Same goes for copying/moving folders and entire folder structures.  

mLindh - the same symptoms can be caused by permissions on other referenced resources as well, such as images, javascript files, etc.  For instance, if you have an image in a sub-site or personal site that is locked down, but is referenced on a page, some users may get this prompt.  (Saw this all the time on SPS2003 with the "Area Contact" picture being locked down in a user's personal site.)  Check your IIS logs, use something like Fiddler, or look for missing images or js/css files to determine what resource is causing the authentication prompt.