The language is strange to us moderns, who have grown unused to philosophical and theological discourse, and who have a cramped and impoverished imagination limited to the material and temporal. The ancients lived in a larger and richer intellectual space than we do today, as they had not limited the world to only what they could perceive with their five senses, as we have largely done. The world was a place full of meaning in its very nature, from its beginning. It contained more than was evident at first glance, and much that was mysterious yet true.
Our author's argument presumes a world full of personal beings, including God, the Son, these beings called angels, and us. At least. From the beginning of the Bible to its end we see beings that are neither us nor God, variously called "seraphim" or "cherubim" or "living beings" or simply "angels" (Gk angeloi) which denotes their function as messengers or representatives of God. We are told little about them, which is not surprising if the Bible is concerned primarily with the relationship between humans and God. We see them incidentally, as it were, enough to tell us that we are not the only sentient creations of God. We are told our own story, and not theirs. If they were only ever called "angels" (messengers) we might conclude that they were only manifestations of God, ways that He himself appears to mankind, as in the case of the three "men" who appear to Abram on their way to Sodom. But in many places they are named as regards "type" (cherubim and seraphim) and even as individuals (Gabriel and Michael). They clearly can manifest themselves in the domain of our own experience (matter/spacetime) but somehow transcend it, as they come and go in ways that we cannot.
In a world full of personal beings of various kinds, one would want to know how these beings relate to one another, especially in considering the place of this mysterious being, "the Son". Our author makes it clear that the Son is not one of the angels, who are servants of God, but a being far above them, having the "glory" of God, being his "very image" and furthermore being the very creator and sustainer of the world. This sounds a lot like God himself, who, if He is anything, is the Creator. Yet the author does not allow a complete identification of this Son with God, insofar as the Son is somehow a separate person sitting at the right hand of God, ruling with Him. Furthermore, the Son is "begotten" by God, and hence has "inherited a name" from God.
Now, one thing that was understood by ancient agrarian civilizations was that like begets like. Cattle beget cattle, sheep beget sheep, humans beget humans. In saying that the Son is "begotten", the author is claiming that the Son is the same type of being as God. This is made clearer in the claim that in being so begotten, the Son has "inherited a name" better than the angels'. We are peculiar today in naming our children according to how well we like the sound of the name. To us, a name is just a sound that "calls" the person. To the ancients, and in the scriptures (which are ancient documents), a Name is much more like the "essence" of a person or thing. This is why God renames certain persons in the scriptures at critical junctures in their lives, when their lives are about to take on additional meaning. Abram becomes Abraham; Sarai becomes Sarah; Jacob becomes Israel. Zacharias is told specifically what to name his son John, and Mary is told what to name Jesus. Adam's naming the animals in Genesis is much more than his thinking up funny sounds to call each one; he is sharing with God in the definition of what kind of things they would be.
So, the "Son" in Hebrews inherits his "name" from God himself, as a naturally begotten son would take on the family name. The names of God, which describe His essential being, are inherited by this Son. Whatever else he may be, the Son bears the names of God.
So here we have the beginning of a mystery: this being called the Son is somehow not exactly the same person as God, yet is the Creator and Sustainer of the universe and the "exact image" of God. He rules with God in the position (right hand) of highest honor; he is the same type of being as God, and shares His name and hence His essential nature. Angels exist, but they are mere messengers, far below this Son.