I think it would not be very beneficial to do two aways at the same place or do research there. Doing two aways basically doubles your chance of pissing someone off and getting black balled. I also don't think that being there an additional month would show them anything new that they didn't get the first time you were there.

Doing research with someone might make a good impression to the 1 person you work with, but the admissions committees are typically made of many people and even if that 1 person loves you (assuming they are on the committee), chances are it won't make much of a difference. Also, if you end up not matching there, you likely will not continue that research.

I would recommend research at your home institution and doing 1 away at the places you think you are most interested in.

Agree with DHT that it's a bad idea. You're better off doing an away elsewhere and getting an additional interview.