Some instructors, particularly in composition classes, will allow you to choose your own topic. If the topic interests you, you are more likely to enjoy your research and to write a good paper. Thus, if your instructor allows you to choose your topic, choose a good one! Don't think that you have to choose one of the "tried and true" topics, such as abortion or the legalization of marijuana, just because you know you will have an easy time finding information on them. After all, even if the information is there, you may not want to read it if the topic bores you. You may be much better off choosing a topic that interests you—even if it is a little quirky.
Example: Your composition instructor has assigned a research paper on a topic of your choosing. You may write on a cultural phenomenon, an historical event, a contemporary issue, or just about anything else that interest you.
Start with your own interests and experience. Consider your hobbies, your intended major, jobs you've had, or situations or problems that you or a friend has experienced. Do you like to listen to music? Perhaps you could write a paper on file-sharing. Are you planning to study nursing? Consider writing on an issue in the medical field. Are you into sports, fantasy, books, dance, shopping, videogames? Consider writing on the reasons why people are attracted to these things or their impact on children.