Twice over the last two days I have gotten requests to change the domain of a web hosting account in cPanel. It’s enough steps that it’s not simple to tell someone how to do, and the fact is only server admins can do it. You need access to the Web Hosting Manager (WHM-the host of the hosts) interface to do it, so this tutorial is really for folks at various institutions (more than 30 now!) running their own Domain of One’s Own package, as well as anyone who happens upon it. Welcome!
For the purposes of this tutorial you will need access to WHM (the management console for a cPanel web hosting server) and WHMCS (the client and billing management system that sits on top of cPanel).
You can make the below changes in WHM or WHMCS first, the order you choose to do it will not effect the outcome.
WHM
I start with WHM, once I am in there I use the quick search tool to find the List Accounts section. From there I click on List Accounts and search for my domain.
Once I search my domain the web hosting account I need to change the domain of will be filtered.
Next step is to click on the more options area (a + icon) to get the various options for modifying this account. The one I need to change the domain of the account is Modify Account.
From there I will be taken to a screen with many of the account’s settings, including the first one: Primary Domain.
Change this field to the domain you want, in my case jimgroom.sites.grinnell.edu rather than jgroom.sites.grinnell.edu
After that change the changes, and be sure to select the “Keep this account on package ‘default'” -even though it is not recommended-and click Proceed. 
After that you should be all done in WHM. It is important to keep in mind that when you change the domain for the site applications you have installed in the hosting account may still need to be updated manually. This goes for everything from static HTML files to databases, changing the domain of the account WILL NOT change the domain in the links of files and databases for any applications installed.
WHMCS
That’s all you have to do with WHM, and WHMCS is just as simple. You login and go to view/search clients and fine the account for the domain you want to modify.
Once you do you click on the number next to the name to view the client’s account.
From there you want to click on the Products/Services tab in the client’s account.
From there you will see the Domain field with the old domain.
Change this to the appropriate domain and click save. 
After that, your account should have a new domain, and you may (or may not) need to make the appropriate changes in any applications currently running on your hosting account.
This rocks. Thanks Jim!