You set the sender name for iCloud email sent from your Mac by going to icloud.com, signing into your account, opening Mail, clicking the gear shaped icon on the bottom left and choosing Preferences, going to the Accounts tab and entering the name you want to use in the Full Name field and clicking Done. Then quit Mail on your Mac and re-open it and you sending name should appear in the drop-down list when you compose a new email.
I'm not sure if that is you problem, but you might check to see if "MobleMe" was somehow listed in the Full Name field in your Mail preferences on icloud.com.