There is no name stored in your private key. All UIDs are bound to the primary public/private key pair (by a special kind of signature). Add a new UID, make it primary and revoke the old one - you're done.
$ gpg --edit-key [key-id]
gpg> adduid # Go through user ID assistant
gpg> list # Get list of UIDs
gpg> [n] # The number of the just added UID, probably "2"
gpg> primary # Set primary UID
gpg> [m] # The number of the old UID
gpg> revuid # Revoke that UID
gpg> save
Be aware that all incoming signatures (if you had any) point to the old UID, not the key; so you will be losing them anyway.