From this post from the Business Catalyst forum:

Login the into the admin first

After login click on the Partner Portal link on the top.

Click on clients tab

Click the site that you want to delete

Under Details section on the righthand side you will find [Delete site] option at the bottom right corner of details section or next to the Upgrade site button.

Click Delete site button a feedback will be asked fill it if you want or just press the Delete site button at the bottom of the form and your site will be deleted

Hope this helps!

EDIT: Here's a screenshot of what this function looks like, found in a very recent BC blog post:

Delete paid and Creative Cloud sites from the Admin Console

We moved the "Delete Site" functionality to the Admin Console of the website in order to help Creative Cloud users and beginners cancel an existing paid website that they don't need anymore. Here's a snapshot of this new feature.

Note that this "new" BC does not go live until July 2nd, so this feature may not be accessible until then.

It looks like the BC console got switched around on me! I've yet to find a direct way to delete a site through BC, but for now try this work-around:

1. Login to the BC Partner Portal with your Creative Cloud account ID and password: http://www.businesscatalyst.com/Admin/Index.aspx?to=PartnerPortal

2. In the top tab bar, click on the "Clients" tab:

3. You should see your sites listed. Under the Actions column for the site you'd like to delete, click on "Details":

4. In the Details panel, click on "[Delete Site]":

Once you click that, you'll be prompted to answer a few questions about why you're deleting your site.

I appologize for not being able to provide you with a more direct answer, but I'll continue to do some research. If I do find out more information about this problem, I'll be sure to post here again.

Cheers!