In Apple's Mail program, when you add an e-mail account you can specify a full name to associate with it, which will be used instead of the account's log-in name or address when sending or receiving messages, and which can make various To and From fields a bit easier to make sense of. While this option is convenient, with Apple's new iCloud service you may find your display name has been changed to your account name or another alias with no apparent way to change it.
If you go to the Accounts section of your Mail preferences, select iCloud, and then click the Account Information tab, you will see a field for your Full Name that is usually editable; however, with iCloud accounts this field may be grayed out so you cannot change it. In most cases the field should contain your preferred name, but in some cases the field may contain your abbreviated account name instead.
Again, this will mainly affect how your name is displayed on received and sent messages, which while purely aesthetic might be frustrating, especially if other accounts you use show your preferred name appropriately so that your messages show a mix of names like "jsmith" and "John Smith" (should those be your log-in and full names).
To change your displayed iCloud full name in this case, you can't use Mail's preferences or even the iCloud settings in the OS X system preferences. Only Apple's iCloud Web services will work. First log into iCloud.com, then click the Mail icon to open the Webmail application.
In the Webmail application, click the gear at the top right and choose "Preferences" from the menu that appears, and you should see a window that appears with settings specific to your e-mail account. Click the Accounts section and you should see editable fields for Description and Full Name. Fill in your desired full name to display, and save the settings. The changes should now be reflected in your iCloud account in Mail on your Mac.