After a few responses of people attempting this, it would seem that this is only fully working on EU servers. A large number of people on US cannot get it to work. If you've gotten this to work on US servers, please let me know if you did it any differently than how this guide describes.
Found this myself a few days ago by accident. If you go to the Faction Transfer service, choose your selected character and put in credit card info but enter the wrong security code, it'll still say the transfer is pending for your selected character and will progress once correct credit card info is submitted. Now, if you log in to WoW, your character will have the normal name and random number after it. Do NOT ticket this as a GM will simply take the letters away. Instead, get peoples attention. Talk in trade, do BGs/LFGs. Eventually you'll get those fags who're jealous of your new badass name and report you to a GM. After the GM realises how this error occured, your name will be flagged for change when you next log in.
Thanks to user 'pnrxt', it has also been confirmed you MUST have a balance of €25+ on the credit card you're using or it won't work. You won't be charged this amount, of course, it's still free.
He has also provided a quicker method of getting your name flagged for change and less likely to catch a GMs attention. Lets say your name is 'Example'. After doing this exploit, your name changes to 'Example2746' and you're now waiting to be flagged for name change. Create a new lv1 character and also name it 'Example', now log in on the character you want changed, in this case being 'Example2746' and you will instantly be flagged for name change. Be sure to delete the lv1 after. Thanks to pnrxt for sharing this information.
I didn't think to take screenshots because this was totally unintentional and, typically, I went back to do it again for more screenshots but...
For now, the only one I have is getting trades attention.
I'll update the thread once the Faction Transfer service is back up on the EU site and post more screenshots when I get them simply for proof. Should be easy enough to follow without them for now.
EDIT: Account management now seems to be back up and I've tried it again. It would seem you need to put in the correct credit card info but the wrong Security Code. Changed info above to reflect this.
I've also discovered another bug while taking more screenshots as proof this works. I did it on two characters to be certain, and it would seem the mage I did it to duplicated. See bottom screenshot.
Be sure to choose Character Faction Change on Battle.net
Then keep ALL information, i.e, Full Name, Email address, Card Number and Expiery date exactly as it is on the card, then change the Security code on the back of your card by one digit. It should still be three digits long.
This is what I get after inputting the incorrect Security Code.
After you've gotten your new name, go to the Faction Change Transactions page. There should be a tab called "Faction Changes Requiring Payment". Just click cancel transaction to ease off suspicion.
This link is for EU servers so it'll vary on US. Keeping in mind US seems to be having trouble with this exploit in the first place.