I'm wondering if it's possible to change the name on the battle net account.

5-6 years ago when I began with WoW we signed the game in my fathers name, but now I wan't to change it to my own name.

I took a look at the Edit Address part at battle net settings but I can't change it.

Hi,

If you need to change details you will need to discuss the matter with our Account and Billing department as they may be able to change the account name if you can prove that you where under-age at the time.

This department can be contacted either by telephone or by e-mail, details of both can be found here:
http://eu.blizzard.com/support/article.xml?locale=en_GB&articleId=23210