The Name format option lets you specify how names display in the Gmail interface, for example in the Inbox and during auto-complete.
You can choose either First Last (e.g., Bob Smith), or Last, First (Smith, Bob), whichever more closely mirrors the culture of your organization.
For example, if your users tend to address one another by first name, use the First Last option.
If your users tend to address one another more formally, using the family name, then use the Last, First option.
You can also allow your users change their display names in their own email accounts. To set this user-defined option, select Allow users to customize this setting. Once you set the option, Bob Smith can log into his email account and change his display name to Bobby Smith if he wants. However, this option does not allow Bob to change how any other users names are displayed; Bob can only change his own display name in his email account.
Please also note name format preferences aren't applied to the Contacts Picker in Gmail. In the Contacts Picker, contacts will be shown as "First name, Last name" regardless of your domain's name format preferences.