How to change the display name next to the email address
The name that appears next to the email address is called the Sender info and it can be changed in your Signatures window. The only case where the display name is not customizable is for Office 365 sync channels, as they do not support setting a different name and can only use the default account name.
To update your signature, go into your Settings, and select Signatures. You can update your Sender info in a current signature by clicking on it, or create a new one by clicking Add a signature.
Type your name in the Sender info section.
This is how a name and email address appear when the Sender info is empty:
This is how a name and email address appear when the Sender info is "Bruce Wayne":
Bruce Wayne <email@example.com>