I think the answer is infinitely big, as it involves a big question on the level of citizens' participation in government, if democracy can be taught, and if Protagoras and Aristotle had some points against Socrates and Plato etc.
Until we solve these issues, we can see user-centered design to help government in various levels. For example,
- Monitoring gov: Designing to monitor government, so that citizens can report issues or request changes. E.g., Open spending.
- Design for citizens: Governments can suggest design guidelines as to help citizens use public services. E.g., Gov.UK Design Principles.
- Design with citizens: Governments or other organisations can start initiatives that help citizens participate in their communities. E.g., Participatory budgeting.
These are meant to be examples and surely not a definite list, more like a starting point I would say. Moreover, I do not necessarily endorse these specific websites but I most certainly applaud the effort.
A mix of those techniques is sometimes put in practice at various hackathons, where citizens' or other organisations provide some ideas/requirements and designers/developers work on these ideas intensively for a short period of time (anything from a couple of days to a couple of weeks). They're exciting to look at, but only UCD in the sense that the developers don't build their own apps and that ideas come from citizens. Because they're brief though, these apps don't go through thorough user evaluation more often than not.