Take a company that you like or know something about and pick one of its products or services. Try brainstorming a few ideas of graphic art logos that you could make for that product or service. This will allow you to use a pre-existing item or entity to inspire you and will give you parameters of what you are working with.
Take an old logo of a product or service and modernize it. This is more difficult because you will have to maintain the message of the original logo while still making it more streamline. Making an older logo sleek and new is an important aspect of graphic design.
Create a product yourself. This is particularly challenging because it causes you to have to come up with both the product and the logo or banner to endorse it. Of course, this also gives you complete creative freedom to make absolutely anything that you want. Make the product something that you would like or something that you can relate to. In doing this, you can better interpret how the logo should feel or what exactly it should portray.
Select a movie that you like or have seen in the past and create a poster for it. Draw images from its key scenes. Compile those images to make a poster that will make someone want to watch the movie.