"A Google Apps user has two names: a display name consisting of a first name and last name, and a user name that appears before the @ sign in his or her primary email address. You can change either or both of these names.
Changing a user's display name changes the way the name appears in the Gmail interface, for example in the contacts list and during auto-complete.
Changing a user's user name changes his or her primary email address, which Google Apps uses to identify the user. Google Apps uses the primary email address as the unique identifier for a user's account: it is the name the user enters to sign into Google Apps, to sync with his or her mobile device, and to share Google Docs and Sites.
If a user simply needs an alternative email address, consider creating an email alias for the user instead of changing his or her user name and primary email address. "
Rename a user - Google Apps Help
Rename a user - Google Apps Help: