To change your Fortnite name, you must change your Epic Games Display Name. Open a browser, go to epicgames.com, sign in, then go to Account and Account Info. Find Display Name, click Edit, type your new name, confirm, and save. Restart Fortnite to see it update.
If the name is taken, add a dot, underscore, extra letter, or small number. On console, make sure you are changing the Epic display name, not your PSN or Xbox name.