Navigate to Site->My Websites and select the website you wish to copy. Once selected, you will open the website's record. In the top right-hand corner of Brick River, click on the more drop-down menu and select copy site.
Here is what you'll see after pressing "copy site"
You need to give the website being copied a new name and short name. You also have the option to copy only the website or the website including all the layouts it uses. It will copy all the existing layouts to new ones and link all the pages in the new site to the new layouts. After you fill out the information and press copy website, a new site will be created and listed under the Site tab in Brick River.