Do an in depth study of a particular artist. While you should, of course, choose someone you are interested in, you should also make sure it is someone that the instructor you are asking to oversee the independent study is familiar with. For example, you should not do a study of Nicolas Poussin with an expert on Impressionism. Your biography should be more than just a chronological listing of the events in the artist's life; it should examine his development as an artist over time.
Another approach to a written independent study in art is to study a particular genre of art, such as the impressionists or pop art. In this case, you would review the history of the genre, looking at the work of its most prominent exponents. You should also look at the influences of other genres on the one you are studying, as well as those that might have grown out of it.
Do a video documentary about the holdings of your local museum. This can be either the university's own collection, or it can be of the museum in your city. The purpose of the video would be to showcase the collection, or at least the highlights. You can yourself narrate the price, interspersing your comments with observations about particular pieces by artists or professors who agree to donate a few minutes of their time.
Conduct a study of the art on display in your local museum, both what is on display and what is in storage. The purpose of this study will be to determine just how decayed or damaged the collection has become. The results of this study could then be used by the directors of the museum as evidence that they need further funding to carry out proper preservation.