If there is a connection, and I cannot stress this if enough, all it would indicate is that yet another biblical myth is based on older stories. It certainly doesn't in any way "prove" that science is "wrong" or that the Genesis is "right" - there are 93 other inconsistencies & contradictions in Genesis. Also, Genesis does not conform to what we know about, for example, the formation of the universe.
Now unto why your hypothesis is extremely unlikely to be viable.
First, the names Adam and Eannatum have no similar meaning or etymology:
- The biblical account of the creation of male and female humankind shows that the man was made from elements of the earth (in Hebrew, adamah, “ground, land”). That's the root of the name Adam.
- Eannatum (𒂍𒀭𒈾𒁺) means "Befitting the E-anna (temple)"
Second, we know when the Jews wrote their creation myth:
- historians traced the origin of the story to the Babylonian exile, when King Nebuchadnezzar II deported the Judahites to Babylonia, where they were essentially held prisoner.
- Nebuchadnezzar II lived two millennia after Eannatum.
Third, we know the myths from which the Jews copied theirs:
In the middle of the nineteenth century, archaeologists were digging in the library of King Ashurbanipal (668-627 B.C.) in the ancient city of Nineveh. They discovered thousands of clay tablets written in a language that came to be known as Akkadian (a distant and much older cousin to Hebrew). One of those texts bore striking similarities to Genesis 1.
Found among the ruins was a Babylonian creation story referred to today as Enuma Elish. It is a story about a highly dysfunctional divine family engaged in a major power struggle at the dawn of time. The heart of the story is where the god Marduk kills his nemesis Tiamat and then fillets her body in two, making the sky out of one half and the earth out of the other. Thus, Marduk claims the throne as the high god in the pantheon.
Scholars have termed Enuma Elish the “Babylonian Genesis.” The reason is that both stories share some concepts that were immediately apparent.
- In both stories, matter exists when creation begins. Similar to Enuma Elish, Genesis 1 describes God ordering chaos, not creating something out of nothing.
- Darkness precedes the creative acts.
- In Enuma Elish the symbol of chaos is the goddess Tiamat who personifies the sea. Genesis refers to the “deep.” The Hebrew word is tehom, which is linguistically related to Tiamat.
- In both stories, light exists before the creation of the sun, moon, and stars.
- In both stories, there is a division of the waters above and below, with a barrier holding back the upper waters.
- The sequence of creation is similar, including the division of waters, dry land, luminaries, and humanity, all followed by rest.
Genesis is just a later Hebrew version of this older Babylonian story, whilst also "borrowing" from and the far older Atrahasis and Epic of Gilgamesh.
Gilgamesh’s possession of a plant of immortality is thwarted by a serpent (compare Gen 3), he wrestles in the night with a divinely appointed assailant who proclaims the hero’s identity and predicts that he will prevail over all others (compare Gen 32:23-32), and he is taught that the greatest response to mortality is to live life in appreciation of those things which make us truly human (compare Eccl 9:7-10).
In the Epic, the gods create Enkidu, who runs wild with the animals in the open country, as a companion for Gilgamesh. There are particularly interesting similarities between the Garden of Eden story in Genesis and the story of Enkidu’s movement from nature to culture and civilization. In both stories, a woman is responsible for the transition of a man who had once eaten and drunk with the animals to a state of estrangement from nature. Once Enkidu is rejected by the animal world, the woman Shamhat gives him clothing and teaches him to drink beer and eat bread—all technological developments that separate humans from animals.
In Genesis, once Adam has eaten the fruit of the tree of knowledge, he covers his nudity and is sentenced to a life of cultivating food by harsh labor. This is the cost of divine knowledge. In Gilgamesh, when Enkidu becomes estranged from the animals, Shamhat tells him that he has become “like a god.” Later, on his deathbed, Enkidu laments his removal from a state of nature, only to be reminded by the god Shamash that while civilized life is more fraught with difficulty and the knowledge of one’s own mortality, it is a worthwhile price for cultural knowledge and awareness.