I would be interested in seeing something like a Department of Efficiency and Bureaucratic Management that would deal with organizing, computerization, and communication between different governmental bodies. Like a bureaucracy for the bureaucracy, if you will. I'm currently living in the United States, but in my home of Hong Kong there is a dedicated efficiency unit, although I'm not really sure of its extent. Of course, I can see possible problems with corruption and partisan mismanagement coming into play, but the concept divorced from reality would be incredibly useful.
Turns out we've already got it. Wish they had more teeth.
A Gross National Happiness committee, like in Bhutan.
That's a tough one. I think it might be nice to have an office that was charged with ensuring that citizen's rights weren't being infringed upon. I think their powers could include fast-tracking issues to the Supreme Court, and also putting a stay on new questionable legislation until it the courts made a determination. However, these entities always seem to have unintended consequences. Personally, I think most new departments that we envision would be to deal with corruption and disenfranchisement. If lobbying and campaign contributions were drastically limited, I think we'd get results that we would hope these new departments might achieve.
With a large enough population, everyone will get their toes stepped on at one point or another.