Can I add my middle initial to my account name ?

If you are trying to change the name that is part of your Apple ID information, log into the link below, and in the first section select Edit. You can change the name there.

That doesn't affect what is displayed as your Apple ID or your user name in the Support community.

Apple ID - Manage          

Your AppleID, and thus the login ID for all Apple services, is an email address.

Your user name here is not an account name, it’s just a forum nick name. Your account name here is the same as for all Apple service accounts - that is, your AppleID email login ID.

And if you mean your @icloud.com email address, that is not editable once created.