How important are independent undergrad projects in grad school applications for CS?
The short answer, at least in my opinion, is definitely to do the independent project if you're considering graduate school. But an important consideration is making sure it contributes to a strong application package.

If you chose to take the additional classes, their grades would be reflected on your transcript and in your GPA, and so there would be an objective measure of how you had mastered their content and in so doing had become a better candidate for admission. 

If you do the independent project, you want to consider carefully how its results will be communicated so as to strengthen your application package. You might at some point discuss this with the project supervisor. If publishable output is a possibility, that's great, but I'm assuming that's unlikely. But if things go well (AFTER it's established that things are going well), you should approach your supervisor about providing a reference letter that you can include with your applications. You don't want to just have that project listed in your application alongside all your other classes; you want to be able to demonstrate how doing independent research as an undergrad made you more likely to succeed as a graduate student (thereby making it easier for the admissions committees to justify offering you a place in their incoming class).