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Does anyone have some insights into which stats or applied math departments have some good information theory researchers? I've seen that Wharton stats and Stanford stats have some people, but it's a difficult topic to try to figure out because Google pretty much just brings up electrical engineering departments.

If you want to go to Chicago or NYU or similar, I'll second the math requirements. In fact, you'd probably be better served by an MS in math than in finance (a large number of the students at top schools have this), and it should be at the best school you can get into. If you do an MS in finance, it should be a research heavy program - very preferably with a thesis component.

Even if you choose finance, you should try to get as many math and CS classes as possible. If you want to work in industry the great jobs go to quantitatively savvy people, and if you decide to get a PhD the quant classes are a necessity. Also, the CS and especially math classes well help you get into a good master's program for finance. You should apply to the programs that interest you. An MS in finance from a top school will be just as good as an MS in fin eng.