This is a thought provoking question.
I know of no firm metrics (other than voting patterns, which are flawed as participation rates are so low 38% - 62% typically for national elections), though there is a plethora of writing on the effectiveness of government. For instance, you can uses STEPS:
Characteristic - Definition
Strength - Relative influence over other stakeholders in the world.
Transparency - Degree of openness in government communication and data availability.
Economics - Soundness of financial footing. Judgment of whether the value delivered is worth the cost of delivering.
Public Goods - Products and services provided by the government to improve quality of life.
Social Goods - Products and services provided by the government to promote social, procedural, and economic justice.
Source - Gov't Executive
The World Bank has a publication on Governmental Effectiveness: Governance Scores
The overall effectiveness of Government ideas might help refine ideas for metrics of a component of the government - e.g. Congress. That's a way to back into the question.
Otherwise you might evaluate along a number of 2 dimensional criteria - reminiscent of the "Guns vs Butter" dialectic in economics.
Use the constitution and Bill of Rights to come up with criteria:
Free Exercise of Religion
Freedom of Speech
Freedom of the Press
Freedom of Assembly
Availability/support for petitioning for the redress of Grievances
Fifth amendment right indices: How many are "deprived of life, liberty, or property, without due process of law?" or does the congress allow "private property be taken for public use, without just compensation"?
Things along this line. This actually seems a very interesting line of discussion.