With the release of Exchange Server 2013 CU10, Microsoft redesigned the Public Folder Picker \ Menu for OWA.
The screenshots below shows the old menu before CU10.
This Is the old Picker
After the Install the right Click menu looks the same
The big change Is In the selection menu below
As you can see the new look is much better.
Folders that can be added to the Favourites are show in bold and folders that cannot are greyed out.