How can I change or add a profile picture or my name in the game?

Facebook:

The name and profile picture is information that is pulled directly from Facebook, so they will be automatically added to the game if you connect to your Facebook account. 

If you want to change either one in the game, you can change them in your Facebook account and it will be updated to the game, too.

You can connect to Facebook in the in-game Menu Settings > My Account.

Apple Sign In:

If you have linked the game to your Apple ID via Apple Sign In, your name and profile picture can be pulled directly from your Apple account. Changing it in your Apple account will change it also in the game.

You can connect to your Apple ID in the in-game Menu Settings > My Account.

Randomized name and Best Fiends avatar: 

You also have the option of choosing a randomized name and a Best Fiends related avatar. You can do this in Menu Settings > My Account > Set Name or Set Avatar.