Thanks everyone for the input, it all helps a lot! I am really shooting for a top 10 in either polisci or economics, so I've just been bugging a lot of people over what I should shoot for on the application.
Just some comments and clarifications:
1) When I mentioned 'research interest' in the original post, I mainly meant how specific I should be in the SOP. I mean, I assume they aren't expecting me to lay out a research proposal for a dissertation, (are they?). And, does having a research interest in a 'hot' field do anything?
2) For the economists: I'll have calculus, linear algebra, ODE and PDE, Probability theory, an intro proof writing course, 1 semester of real analysis, and maybe 1 of abstract algebra (and a math for econ phd course sequence). Will not having that second semester of analysis and algebra hurt my chances? Are there any other specific courses that could help?
3) I forgot another part that might warrant some discussion: does job experience do help? I mainly ask because I have been told that working for a year or two between undergrad and phd is a good experience to have before fully committing to the phd.