Cody Rhodes is free to use his a father's last name. He just chooses not to.
In a Q&A session on Twitter, a fan asked Cody when he would be able to use the last name Rhodes when promoting his matches. Cody revealed that he can do so now, but enjoys going by one word.
When Cody left WWE in May 2016, he wasn't allowed to use the last name Rhodes as it was intellectual property owned by WWE. As such, Cody began going by his first name only when taking Independent bookings. In April 2017, he told Sports Illustrated that he was starting to like having only one name.
“I’ll tell you this, and I’ve not told anyone this, but I don’t mind that WWE took away my last name. Deep down, in my bones, I definitely want it back – and I have plans to get it back – but there is something to being Cody. The longer I don’t have a last name, the more I’m OK with it. That’s not to say WWE is holding it ransom. It’s literally an intellectual property law that easily can be remedied, but there is something about being Cody that I don’t mind." He continued, "There is something to not always reminding people of a show they’ve already seen, but instead embracing the one right in front of him. I am Cody, and I can promise you that the future is going to be even better than the past.”
Despite not being able to legally use the last name Rhodes following his release, Cody has done just fine for himself on the Independent circuit. And the majority of the audience knew who was being talked about when the name Cody was being promoted.