Boyd’s Bible Dictionary:

“Abram the Hebrew”
(Gen. 14:13), that is, eber, the one who had “passed over” the Euphrates, westward. Hence, “seed” or descendants of Abraham. Among themselves, preferably, Israelites, from Genesis 32:28. Jews, that is, Judahites, Judean, after the captivity.

Concise Bible Dictionary:

Designation of Abraham and of his descendants. The name is first met with when Lot had been carried away prisoner, one came and told Abram “the Hebrew” (Gen. 14:13). Hence it is applied to Abraham’s descendants through Isaac and Jacob in distinction to the name of Israelites (from the name of Israel given to Jacob), which is their covenant name, the name of promise. It may be remarked how Saul king of Israel had lost the sense of this when he said “Let the Hebrews hear” (1 Sam. 13:3).
The term occurs in the New Testament only in Acts 6:1 to distinguish the Greek-speaking Jews from those of Palestine, and in 2 Corinthians 11:22 and Philippians 3:5 concerning the ancestors of Paul, wherein, to meet the caviling of the Judaizing teachers, he calls himself a Hebrew of the Hebrews, one who had descended without any Gentile or proselyte blood.
It is not very clear why Abraham was called a Hebrew. It is generally supposed to be derived from his ancestor Eber or Heber; but it will be seen from Genesis 11:17-26 that there were five generations between Eber and Abraham, so by this derivation many others might have been called Hebrews. Genesis 10:21 Says that Shem was “the father of all the children of Eber.” This shows that the Hebrews were Shemites, but many other tribes were “Shemites” that could not be called Hebrews. In scripture the name is not applied to any except to Abraham and his descendants, and only to those who descended through Isaac and Jacob, to the exclusion of the children of Ishmael and Esau. So that there must be some other reason for the name and for its being thus restricted.
The root of the word is “to pass over,” as when one passes over a river, or from one region to another. Abraham was bidden to leave his country and his kindred and to go into the land of Canaan, and the word Hebrew is not employed until Abraham had left his country and was in the land of Canaan (Gen. 14:13). When there he was a “sojourner,” in a strange country, dwelling in tents (Heb. 11:9). The name was therefore characteristic, and the people of the land could go to Abraham the “sojourner” and tell him that Lot had been taken prisoner. Joseph when in Egypt said he had been stolen from “the land of the Hebrews” (Gen. 40:15). The above characteristic was doubtless subsequently lost, and nothing seen in it but the natural descent from Abraham through Isaac and Jacob; the same persons being mostly called Israelites. The descendants of Ishmael and Esau were not sojourners in the promised land, but wandered whither they would. The name Hebrew does not occur in the Old Testament after 1 Samuel except in Jeremiah 34:9,14 and once in Jonah 1:9.