Does hubski have any suggestions?
If you're looking for a nice book to follow, Land of Lisp is one that I have which I'd recommend. You learn about lisp while developing a game, which I know isn't exactly what you're looking for but it's the only thing that came to mind.
GOFAI type projects are a good way to experiment with Lisp, because the language's history and much of the literature before Practical Common Lisp are steeped in that kind of thing. Game playing, computer algebra, theorem proving, that sort of thing. If you really don't like that sort of thing, or want something more practical, tool-building is always fun. Generating Gtk bindings from gobject-instrospection metadata would be useful; generating X bindings from the XCB protocol descriptions probably less so but fun if you're into that sort of thing. Common Lisp's arrays are don't get much attention, but they're really well-thought out. Lisp doesn't have much of a reputation for numerical work, and it won't give you as much opportunity to play with the sorts of things people rave about Lisp for, but there's some cool stuff there you won't see in many blog posts. Writing a raytracer with a DSL for scene descriptions wouldn't be a huge project.
I don't know if this helps, but when I was at University, many years ago, I had a data structures class and decided to use Lisp for the course-work. We could use any language, and most used C or Pascal (C plus-plus was young then, and Python not yet born). The major project was red-black search tree implementations. My code was vastly smaller than that of my peers (but probably not faster), and it was very elegant, but not at all easy to fathom. Others would turn in thousands of lines of code, and mine was rarely more than could be printed on one or two pages. It was an amazing experience. The recursive nature of a lot of data structures lend themselves well to the inherent recursive nature of Lisp and Lisp-like languages.
[ mk, I had to spell out C plus plus. Is there a way to not have the plus signs swallowed by the markup? ]
I found it a little difficult to go back and forth between Lisp and other languages. Expect a bit of mindfuckery unless you dive in to the Lisp world with both feet. It's hard to describe, but the mindset for Lisp doesn't serve you well for languages that aren't so functional-oriented. Python is sort of between those two worlds.