An AD user by name domainName\ADUserName added in your SharePoint site in Site Owner group.
Now you change the user name in AD ,this change will not reflect in Sharepoint.
On reimport of user profile will update the User Profile Database . Sharepoint group uses UserInfo Table
STSADM -o migrateuser -oldlogin domainName\ADUserName -newlogin domainName\ADNewUserName -ignoresidhistory
This command will migrate the user but we need to check the display User Name.
Its better delete and add the User in Site Collection