From the New Project page, click on the link to your saved designs. On the designs page, click on a design's thumbnail, then click Delete.
The design thumbnail, if any, shown on the New Project page can't be deleted. It's just replaced when you start on a new design. It is removed when you logout.
If you just want to work on a new design, you don't have to delete the current one. Just go to the New Project page and start a new one.