I don't see that anyone here has helped you do this, so I'll answer:
You can change the name on the personal information page ->
https://bitz.io/personalInformationBut in order to do this, the site also requires you to fill in the "First name" and "Last Name" fields

At least that's the reason why I decided to keep this random funny name that the site generated for me.
Thanks for sharing the answer.
I will also keep my random username. I usually just change to bitmover, but adding more personal information about me isn't nice for now.
Was anyone able to claim those free btc?
Yep, I just tried it a few minutes ago, it loads for a few seconds like the screenshot you shared, and the claim button pops up. I was able to claim 50 Satoshi. It seems like the amount increases if you collect Free BTC every day.
I just claimed 500 satoshi now, just for verifying my email! Amazing, it is working now!