So I created a test account and did some trial runs and I have found this to be the best method:
(Note: We have Active Directory sync'ing to our Google Apps account using Google Apps Directory Sync.)
For this example we are renaming Robin Scherbatsky to Robin Stinson.
1. Find the user in Active Directory, right click on their account and click "Rename".
2. Type in the user's new last name.
3. The "Rename User" window will open, change the "Last name" field and also update the "User logon name" field.
4. Please note that I have NOT changed the email address yet!
5. Run "Google Apps Directory Sync" and you should get the following output:
Proposed changes:
Delete: 0
Modify: 1
Create: 0
Modify (all proposed changes) - 1 total users affected
Modify user 1: "rscherbatsky"
Change family name to "Stinson"
6. Now update the "E-mail" address field for the user to the new email address:
7. Run "Google Apps Directory Sync" again and you will get the following output:
Proposed changes:
Delete: 0
Modify: 1
Create: 0
Modify (all proposed changes) - 1 total users affected
Modify user 1: "rscherbatsky"
Change primary address to "rstinson"
8. At this point I run "Google Apps Directory Sync" again as sometimes it is not able to perform all the update steps in one sync, after running a sync for the third time, I get the following output:
Proposed changes:
Delete: 0
Modify: 1
Create: 0
Modify (all proposed changes) - 1 total users affected
Modify user 1: "rstinson"
Remove 1 existing aliases
"rscherbatsky"
9. Next, add the user's old email address under the "proxyAddresses" attribute in Active Directory:
10. Run "Google Apps Directory Sync" for the fourth time and you will see the following output:
Proposed changes:
Delete: 0
Modify: 1
Create: 0
Modify (all proposed changes) - 1 total users affected
Modify user 1: "rstinson"
Add 1 new aliases
"rscherbatsky"
11. Now it's time to update the user's home directory, locate the folder, right click it and rename it to match the user's new username:
12. Finally, go back and update the user's "Home folder" path to point to the renamed folder. I use the %username% wildcard, you can also type in the full username.
13. You will be asked to confirm that you want to grant full control to the home folder, click "Yes" and you're done.
At this point I have the user logon to their laptop or computer using their new username. Windows should detect that it is the same account and the profile from their old username will be loaded so their desktop, folders, settings etc... are the same.
Sync Centre also seemed to sync correctly with the new settings without having to clear the offline files cache.