I use cpanel, which has horde, neomail, and squirrelmail pre-installed. I want to give my non-technical clients control over their own email, but making them sign into cpanel to check mail or change their passwords is too difficult.

How do I set up a mail service that allows them to add/delete/edit their own accounts, and set it up so it's accesble from, say, mail.businessname.com?

Setup a mail server on a machine, then set the MX records to point to that machines IP. And then create a new Address (A) record for your domain that points to mail or webmail which will enable you to have mail.businessname.com or webmail.businessname.com