Go to any anime list page and click the get random anime button at the top. After a bit, a dialog box will pop up with a link for you to copy of your random anime.
The script will ignore any shows on the current anime list you're viewing. So you can be on the completed shows page, and it will only ignore shows you've completed. If you're on the all shows page, it will ignore any show on your list.
It works on any anime list page, so you can choose a random anime for your friends!