There are various reasons you might need to change someone's AD account such as someone getting married and change their last name or perhaps someone wants to go by their nickname.
Although it can be a bit of a pain it really is not all that difficult.
Login to your Exchange server and find the users mailbox and go to the "Email Address" tab and select add to add their new email address that reflects the name change. Make sure the new email address is the one that is bolded.
You may need to uncheck the box about "Email address policy".
Login to your Domain Controller and find the user you want to change. Right click and select "Rename" this will bring up a new windows where you can change all the relevant details. Make all the necessary changes to reflect the new name.
You will want to be able to see the attribute editor, in AD Users and Computer select "View" then "Advanced Features" Then be sure to open up the users properties from the OU and not by searching for it. You should now see the attribute editor in the users properties. Be sure to check that displayname, mail, mailNickname match the new user-name.
Lastly you will want to get the user to log off and then log back on with their new username. One thing that won't have changed is the user's folder on their computer.
One option would be to backup the users profile and then delete the profile and have them log in. This will recreate the profile with the correct name and be linked properly. This is a bit invasive as it does require a lot of recovery to get it back to the way it was.
A much easier way is to login to the PC with an admin account and rename the users profile folder to match the new username. Then open Regedit and go to:
Click on the SID that is for that user if there is more than one. Example: S-1-5-21-2944774474-1080414133-2956492554-1001
Find the ProfileImagePath and double click it to open, now change the "Value Data" to match the new username.