I have the client's iPad in hand and can provide more details now. Everything is working fine logged in as that same user from a desktop computer and from a Samsung tablet here in my office. The client says his father's iPad also works fine.
The amount of free disk space does not seem to be an issue. This is a 16GB model with 9 GB free. Version 7.1.2(11D257) Model MD534C/A. Using WIFI, as are the other devices we tested in the office.
Inbox is highlighted, but the status bar in the main window says "Mailbox is empty"
Compose button is greyed out (along with all other buttons)
Tapping on any other folder flashes for a second, but does not change to that folder.
Settings, Address Book and Mail buttons in top right are not selectable from the mail screen
Manage folders in bottom left can be selected and does bring up the Folders screen.
From there you can access all of the Settings screen except Preferences. Drill down to Filters/ManageSieve/Vacation works.
Address book was selectable from the Settings/Filter screen, but not from Settings/Folders. I know this sounds crazy but it is quite repeatable. Going back to test, the Mail button also worked from the Filter screen, but not the Settings button.
Tapping Personal Addresses does not bring up the contacts in the Address Book. There is a status message at the bottom which just says "Loading..."
There are no errors in the web server or dovecot logs.