Hi Fedrica,
You could create a copy of the existing Facebook Connect Logon web part and modify it to fetch the user's username and full name using the
Facebook Graph API. Once you have the information, you can update the user profile with the details.
Creating a copy of the web-part is recommended over modifying the existing web-part as otherwise your changes may be over-written when you apply hot fixes or upgrade.
Hope this answers your question.
Yasser Abbasi from MMT Digital's K-Team
Kentico Certified Developer