Your display name has to be same as your username.
There are some users that have different display name than their username, but theymos doesn't accept such requests from any user.
If requests are approved by theymos, he can change display name to a different one than username but not easily to get approval from theymos.
A new user created an account and quickly no longer likes that username is not a reason to force theymos doing this change. So an easier choice for OP is abandon this account and create a new one with a username he likes.
Each user has a display name, which is the only name ordinary users see, and a username, which is what you use to log-in with. Almost always, they are the same.
If I change a name for reasons of appearance, then I only change the display name. The username remains reserved, the user can still login using the username, PMs can still be sent to that username, "search member" allows searching for the old username, etc.
If I change a name for reasons of privacy, then I change both the username and the display name. In this case, it is possibly hazardous to reserve the old username, since it allows someone to test for the existence of that username, possibly defeating the privacy benefit. If you're worried about being impersonated, then simply don't seek to have your username changed.