Although nobody really expects an undergraduate to be able to propose a specific PhD thesis topic, you should define your general interests and demonstrate your familiarity with current problems in the field. A prospective advisor will be most concerned about your interest in, commitment to, and potential in his or her field. The best way to give your letter and application substance, then, is to have done something in the field as an undergraduate. Ask your college professors if they know of summer opportunities at Bowdoin or elsewhere. You will discover a wide range of opportunities, stipends, and grants at various institutions across the country.
Letters of recommendation can carry a huge amount of weight.
So think carefully about whom you want to write letters on your behalf, get to know them personally, and impress them with your promise. It can be helpful to have one or two of your letters from professional researchers outside of Bowdoin if they know you well. Clearly, your senior research advisor is a key recommender. Such fellowships often rely heavily on undergraduate grades and GRE scores.