What is the connection of Eber and the Hebrew language?

Answer 1

Our tradition states that Hebrew was the language with which God created the world (Rashi commentary, Genesis 2:23, quoting the midrash); and it is the language in which He spoke on Mount Sinai.

Since it was considered a holy language and was used for prayer and the teaching of religious tradition, it was not spoken in mundane contexts and wasn't taught to just anyone. It was handed down from individual teachers to disciples as part of the original tradition; and the same goes for the art of writing (letters on parchment, as opposed to cuneiform or heiroglyphics).

After the Flood, the Hebrew language had a brief period in which it was generally known, thanks to Noah (see Rashi commentary on Genesis 11:1). This is why many hundreds of Hebrew words have cognates in languages as diverse as German and Japanese.

After the Flood also, the knowledge of Hebrew eventually declined (see Genesis ch.11) and was preserved only among the Western Semites, the ancestors and cousins of Abraham. Eber, from whom our word "Hebrew" (Ivrit) is named, was a Semitic descendant of Noah and ancestor of Abraham. He was one of the major transmitters of the original traditions. He is credited with having broadened the Hebrew language, which is why it bears his name; and some Hebrew grammatical constructs are attributed to him by certain Jewish researchers.