It appears where, exactly? What screen? I'm trying to figure out where you are seeing this name as opposed to your Apple ID. So I need to know exactly what screen you are seeing it on....
Or are you talking about the email being used for the Apple ID itself? If that is what you are referring to, you can EDIT it to a new email address via Manage your Apple ID. However, before you do that, be SURE to sign out of iCloud on all of your device and Mac. If you do not, it will cause issues. You probably want to sign out of Messages, FaceTime, and iTunes & App store on all of the devices and Mac as well.
Then you can go to: Apple - My Apple ID, sign on with the current Apple ID, and then EDIT it to the new email address you want it to be. Note: you cannot change an Apple ID that is using an Apple domain like @me.com, @mac.com, or @iCloud.com. Any other email address being used for the Apple ID can be EDITed to a new valid email address.
Cheers,
GB