Etymology
The modern English word "Hebrew" is derived from Old French Ebrau, via Latin from the Ancient Greek Ἑβραῖος (hebraîos) and Aramaic 'ibrāy, all ultimately derived from Biblical Hebrew Ivri (עברי), one of several names for the Israelite (Jewish and Samaritan) people (Hebrews).
It is traditionally understood to be an adjective based on the name of Abraham's ancestor, Eber, mentioned in Genesis 10:21.
The name is believed to be based on the Semitic root ʕ-b-r (עבר) meaning "beyond", "other side", "across". [20] Interpretations of the term "Hebrew" generally render its meaning as roughly "from the other side [of the river/desert]"—i.e., an exonym for the inhabitants of the land of Israel and Judah, perhaps from the perspective of Mesopotamia, Phoenicia or Transjordan (with the river referred to being perhaps the Euphrates, Jordan or Litani; or maybe the northern Arabian Desert between Babylonia and Canaan).[21]
One of the earliest references to the language's name as "Ivrit" is found in the prologue to the Book of Ben Sira, from the 2nd century BCE.[23] The Hebrew Bible does not use the term "Hebrew" in reference to the language of the Hebrew people;[24] its later historiography, in the Book of Kings, refers to it as יְהוּדִית Yehudit "Judahite (language)" [2 Kings 18:26].
History
... Scholars debate the degree to which Hebrew was a spoken vernacular in ancient times following the Babylonian exile when the predominant international language in the region was Old Aramaic.
https://en.wikipedia.org/wiki/Hebrew_language#Etymology
Previously
The earliest examples of written Paleo-Hebrew [are said to] date back to the 10th century BCE,[11] ...
... The language was not referred to by the name Hebrew in the Bible, but as Yehudit (transl. 'the language of Judah') or Səpaṯ Kəna'an (transl. "the language of Canaan") [Later Hellenistic writers such as Josephus and [the author of] the Gospel of John used the term Hebraisti to refer to both Hebrew and Aramaic, Sáenz-Badillos (1993)]. Mishnah Gittin 9:8 refers to the language as Ivrit, meaning Hebrew; however, Mishnah Megillah refers to the language as Ashurit, meaning Assyrian, which is derived from the name of the alphabet used; in contrast to Ivrit, meaning the Paleo-Hebrew alphabet [Hoffman, Joel M. In the Beginning : A Short History of the Hebrew Language. NY, NY Univ. Press, 2006, p. 169].