However, they no longer need it and it was being a pain to remove - no matter what option I tried, it just wouldn't go!
Trying to "Close" the mailbox results in the error
"This group of folders is associated with an e-mail account. To remove the account, click the File Tab, and on the info tab, click Account Settings. Select the e-mail account, and then click Remove."
After a while, I found that a solution was to go to the Exchange "account settings", then click "More Settings", go to the "Advanced" tab, and click "Add", choose a different Mailbox where the user has permissions.
This seems to refresh the Additional Mailbox section.
Next, click ok/next etc, and go back to that screen. The list should now be populated with the box you just added and the original box. You can remove both as usual.