I was on a project creating an extranet using Microsoft SharePoint Foundation 2010 and ran into this problem: The users could not edit their own name. This was a bit annoying as they logged in using their Microsoft LiveID account, and this had a name like this: 00003f879q83yd9@live.com

The edit-button was actually just gone:

When I logged on as a Site Collection Administrator he edit-option was visible:

Actually this was because users do not have the correct permissions to the web application. As I did not want the users to have too many rights. I added a new permission policy in Central Administration, Web Applications for the web application:

     

Grant the users that should have this opportunity the permission policy by adding their security group on the web applications User Policy: