A user's Email Address is stored within a profile property. By default a user isn't allowed to change his or her own e-mail address. The email address (Work e-mail) and whether or not user's can change their own can be specified in the settings of the User Profile Service Application.
- (As an administrator) open Central Admin
- Under "Application Management" click "Manage Service Applications"
- Locate and click the link "User Profile Service Application". If this one is not there you haven't created a User Profile Service Application yet.
- Under "People" there are two relevant links here. The first is "Manage User Properties", the second is "Manage User Profiles"
- Under "Manage User Properties" you can specify the settings for the Work e-mail property (i.e. whether or not the user is allowed to edit his or her own e-mail address).
- Under "Manage User Profiles" you can locate a user's profile and edit the e-mail address from there
By default it could take up to one hour to "see" the new e-mail address througout all web applications bound to the user profile service application. A timer job is taking care of this and you could kick this one of to run manually once by going to "Check Job Status" screen in Central Admin (_admin/Timer.aspx). Here locate the job definition with the title "User Profile Service Application - User Profile to SharePoint Full Synchronization", open the "Edit Timer Job Page" by clicking on the Job's name and then click the "Run Now" button.
Hope this helps!