Many people was called by a name after their forefathers such as the descendants of Canaan were called the Canaanites, the descendants of Ishmael were called Ishmaelites, then we see the Moabites, the Ammonites (Gen. 19:37-38). Why was Abram called "the Hebrew" in his time (Gen. 14:13)?
Ivri עברי is a descendant [of] "Ever" עבר the Noachide. - Each עברי Ivri speaks the holy language of עֵבֶר Ever using knowledge from the עָבָר Past which maintains the monotheistic belief in one Elohim from נֹחַ Noach & שֵׁם Shem. | Avram is also described by his actions in [Genesis 12:6] when יּעבר Ya'Avor "[He] Crossed" out the polytheistic society of חָרָן Haran to consciously focus on only the true worship of יְהֹוָה YHVH The God of Noach, Shem & Ever.
See [Radak - Commentary on Genesis 14:13]
העברי a member of the tribe of Ever, he and all his descendants proudly regarded Ever as their ancestor. They were the only people who still spoke Hebrew, the original language of man. Subsequent descendants, other than the line which ran through Avraham and the Israelites, changed their language to Aramaic, and were called Aramim, i.e. members of the people of Aram, in recognition of their specific language. Even Lavan, a third or fourth generation descendant of Ever, was already known as לבן הארמי, Lavan who spoke Aramaic. The tribe called עברים was Yaakov and his offspring.
Abraham is a descendant of Eber (whence we get the term Hebrew) who was a descendant of Shem (whence we get the term Semite) (Genesis 10:21). "Hebrew" therefore simply means 'Eberite,' much in the same wat that "Canaanite" means 'of Canaan.' Given that Abraham is a descendant of Eber, and not the other way around, it's not surprising that people refer to him as an Eberite.