How to Change Riot ID / Summoner Name

The username of an account (what you use to log in) can't be changed. You can only change the name visible to other players (Riot ID).

First, go to:
https://account.riotgames.com

If you are already logged in, open the link in incognito mode, different browser or log out from the other account.

After logging in, you will be redirected to the account management page. Now you just need to input the new name using the form on the screenshot.

Note: You might have to change the tag line as well for it to work. Tag line is the "NA1" on the screenshot.