I would suggest Natural Law. The idea behind natural law is that we can use reason to recognize some values and rights that we have because we are human. One of the most famous expressions of this idea is from the US Declaration of Independence:
When in the Course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume among the powers of the earth, the separate and equal station to which the Laws of Nature and of Nature's God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.
Yes, the Declaration does mention "their Creator" and "Nature's God", which has religious connotations, but the main idea is that because we are human, we have certain rights that we don't have to earn from or be granted by a higher power; we are born with them. We can discern right from wrong using our reason and our observation of Nature. Because of this, human-made laws should be derived from or based in Natural Law.
Plato identified four Cardinal Virtues, prudence, justice, temperance, courage, that have been expanded upon by later philosophers. Justice, or fairness, would be the virtue that corresponds to the Golden Rule mentioned in the question. Some Christian religions, particularly Catholicism, have incorporated Natural Law concepts into their views, which causes some of the overlap between secular law and religious teachings.