Which is the longshot, finding the mentor or the NSF?
My experience is that top departments, and fellowships like NSF, basically want people who are already grad students. Actually, NSF seems to have more to do with the merits of the research problem than the student, per se. I don't think I have a shot without enough research experience to have a rough idea of what "my problem" will be.
An NSF graduate fellowship is always a long-shot. Every good first- or second-year graduate student in the country is applying for one, and the competition is intense.
More to the point: the competition is at such a high level that the outcome rests on things that are far more subjective than you might like. For example, volunteer experience and educational outreach can be decisive factors when all the candidates are equally good on paper.
Make a good story about how you want your work to help other people, and also play up the diversity angle if you have something like an immigrant parent. That's what my NSF friends did (except those who were so excellent that they didn't have to jump through hoops).
The cost you pay for this is that your place in grad school and your fellowship are now predicated on your ability to bullshit. This emotional weight may be worse than just plain not getting the shiniest fellowships, or into the most prestigious school.
Yeah...sometimes that works. But if you and your friends thought of it, what makes you think that every other smart grad student in the country didn't think of it, too? (Except for the "have an immigrant parent" part, of course.)
My experience is that top departments, and fellowships like NSF, basically want people who are already grad students. Actually, NSF seems to have more to do with the merits of the research problem than the student, per se. I don't think I have a shot without enough research experience to have a rough idea of what "my problem" will be.