Open source is all about engaging its users, and that's what government is supposed to do. By using open source software, the government will directly be utilizing any civilian programmer who feels like contributing to the code, be it a patch or a derivative work. Think about the sprawling open source communities in existence, and think about them if the scope was expanded to the national level. Everyone could get involved, and not just by going to the voting booth.
This also includes software for citizens to keep tabs on their government. Projects like Apps For America help propagate open source software for citizens to more easily follow the happenings of politics, including those of their local representatives—created by the people, for the people.