Our Gemara on Amud Aleph discusses the Laws of the Sons of Noah. Some have raised the question: Why are they called the “Laws of the Sons of Noah”? Since these laws obligate all of humanity, shouldn't they be called the “Laws of the Sons of Adam”?
Some offer a simple answer: The only descendants from Adam who remained post-Flood were the Sons of Noah, so the term is technically accurate. Others answer that since Adam was not permitted to eat meat, the prohibition against taking a limb from a live animal did not apply to him. Thus, Adam had six laws, not seven, making the term "Laws of the Sons of Noah" precise.
I would like to suggest a different idea. There are two types of law. One is a natural law—an inherent sense of justice and the will of God. (We will discuss this further on the next daf; see Psychology of the Daf, Sanhedrin 58.) The other type of law is covenantal, stemming from a formal agreement. The mitzvos in the Torah are covenantal; they derive their authority from the covenant at Mount Sinai. However, an earlier set of laws emerged from the covenant with the Sons of Noah (Bereishis 9:12). I believe they are called the Laws of the Sons of Noah because this was the point at which these laws took on the form of a covenantal obligation.
Adam, too, was bound by certain laws—after all, Kayin was punished for murder. But these laws arose from an intuitive moral sense rather than a covenant. By way of metaphor, two strangers have an unspoken social contract with certain expectations—at a minimum, neither would brazenly steal from the other. However, members of a clan or community establish specific rules and expectations tailored to their needs. Likewise, Adam’s children—and all of humanity—were obligated to follow certain laws by virtue of their intrinsic moral nature, whereas Noah’s children were obligated because of the covenant that followed the Flood.
Someone pointed out to me an interesting pattern regarding covenants in the Torah. A covenant seems to be sealed—or at least strengthened—through crisis and redemption. The Torah was given after the Exodus, and God even identified Himself as “the One who took you out of Egypt” (Shemos 20:2). Likewise, the laws given to Noach were established after he emerged from the Ark, having survived the Flood (Bereishis 9:8-17). Even though we should act morally simply because it is proper and aligns with God’s will, the Torah emphasizes the contractual nature of mitzvos. God took us out of Egypt, and therefore we owe Him obedience; God saved Noach, and thus he and his descendants were obligated to follow His commandments. This is why these laws are described as the Laws of the “Sons of Noah” rather than the “Sons of Adam.”
I believe this reflects the Torah’s pragmatic approach to human nature. When encouraging obedience, the Torah does not focus on lofty abstract concepts such as Olam Haba; rather, it speaks in terms of the tangible rewards of a good life and material prosperity (Devarim 11:13, Vayikra 26:3). Similarly, while these laws are inherently just and correct, human beings are often motivated more by a personal sense of obligation than by abstract ideals. We obey because we feel a reciprocal relationship with God. Abstract concepts alone do not motivate most people as much as a personal connection, and much of the Torah’s imagery—as well as the language of our prayers—is designed to foster this sense of relationship with God. (See Rambam, Yesodei HaTorah 1-2.)
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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Rabbi Simcha Feuerman, Rabbi Simcha Feuerman, LCSW-R, DHL is a psychotherapist who works with high conflict couples and families as well male sexual health. He can be reached via email at simchafeuerman@gmail.com