To begin with, "Ivri" seems related to the biblical Eber, grandson of Noah through Shem. After all, they are from the same root and all Ivrim are descendents of Eber. However, not all descendents of Eber are Ivrim. Furthermore, the Torah calls descendents of Eber something else: benei ever (see: Gen. 10:21). It would thus not seem that an adequate definition of 'Ivri' is "one who is descended from Eber".
Another suggestion is that Avraham was designated 'Ivri' because he was one who "crossed over the [Euphrates] River", "me-ever ha-nahar" (Avraham is so referred in Joshua 24:3). The question then arises as to why this term should apply to Jewsisraeliteswhatever such a very long time later, particularly since a) as we shall see, not all descendents of Avraham are called Ivrim and b) early appearances of 'Ivri' in Tanakh are from the perspective of Egyptians who, it would seem, wouldn't think of the Euphrates as "the river, over". For Egyptians "the river" meant nothing but the Nile.
At this point its worth mentioning the well-known and well-established point that in the Bible Jews, Israelites--whatever you wanna call 'em--are generally seen as Ivrim (Hebrews) from the perspective of outsiders. Engelmayer notes this (as well as the theory linking Ivrim with the Hapiru, something he [and I] consider tenuous).
But he goes further. How, he asks, can Hebrew have been an ethnic designation if it only consisted of a few people at first? The answer is, presumably, that it wasn't an ethnic designation at first. If so, what was it?
Examining the texts, Engelmayer came to the conclusion that if there is an identifiable characteristic linking all who are called 'Ivri'--that is to say, something that Avraham, Yitzhaq and Ya'aqov were, but Yishmael and Esau were not--it is this: a Hebrew is 1) the progeny of family members, 2) who themselves were products of unions among family members who, in turn, 3) married family members. Got that? Again, an Ivri is someone from a specific family through both parents who were themselves from the same family by both parents, who marry other members of this same family. (a qualification will be added to this definition soon)
Yishmael's mother was Hagar, who was not part of this family, so he is disqualified. Furthermore, his wives were not from this family. Avraham, on the other hand, married Sarah who was from this, his own family. Yitzhaq was the son of Avraham and Sarah, both members of this family, and he too married into this, his own family. His sons were Hebrews too, or rather one of them: Ya'aqov married into this family as well. Esav didn't and consequently wasn't a Hebrew.
Let's look at the evidence. Just recently we read how concerned Avraham was that Yitzhaq marry into only his own family. Yishmael? The benei Keturah? That really wasn't his concern at all. As their mothers were not Hebrews, it wasn't crucial according to whatever value system there evidently was in this family of Hebrews, that they too marry Hebrews.
Esav, on the other hand, is a huge disappointment to his parents over his choice of marriage partners; he married Hittite women (Gen. 25:35). Why? Because Esav could have and should have retained his Hebrew status by taking a Hebrew wife, one that could only be obtained at this point in time back in "the old country". Ya'aqov did that. Esav didn't-- in fact, at one point Esav realized that he screwed up and tried to rectify it by marrying a daughter of Yishmael. Apparently Esav realized that his parents placed a premium on not marrying local women and marrying in the family, so he did (Gen. 28:6). Trouble is that it wasn't merely a matter of marrying kin or countrymen (namely just anyone back in Paddan-Aram). It was a matter of marrying only another who was considered an Ivri.
And now we run into a problem. Ya'aqov had children by four women, only two of whom were Hebrews. The others, Bilhah and Zilpah, were mothers of four sons of Ya'aqov. Furthermore, at least three others of Ya'aqov's sons married women who were not Ivrim, Shim'on, Yehudah and Yoseph. That leaves five who may have (the Torah doesn't actually say who the others married).
Yet Yoseph is specifically regarded as a Hebrew by the Egyptians. This isn't troublesome, as before he married an Egyptian there was no reason, according to this definition, why Egyptians wouldn't have regarded him as an Ivri. Presumably this is how Yoseph self-identified, whether or not it is correct that an Ivri is as defined above. In Tanakh, Jewsisraeliteswhatever, are found identifiying as Hebrews among outsiders. This is similar to how some people call Jews "Yidden" among insiders but "Jews" for outside ears.
What about all the brothers, though, who are regarded as Hebrews by the Egyptians (Gen. 43:32)? That shouldn't be a problem either. From the vantage point of outsiders, brothers of Hebrews could well have been Hebrews regardless of technicalities. By the way, this peshat makes what might be considered the radical case that, at least in the early part of Jewish history, not all Jewsisraeliteswhatever, were considered Hebrews. In this case, only four of the brothers would have actually been Hebrews while six others were technically not Hebrews.
This begs yet more questions. If Hebrew meant something so narrow then how did the Egyptians come to know the term and use it in the first place? What about Yoseph's claim that he came from "the land of the Hebrews"? That certainly sounds like the term represented a geographical locale. Yet at this point the Hebrews in Canaan were but a handful, even allowing for the extended family that was with Ya'aqov and not specifically enumerated in the Torah. How would the Egyptians have even understoof what Yoseph meant? It would be like a Rhode Islander implying that his "land of Rhode Island" refers to North America in general. But the Egyptians knew what he meant.
The answer might be that from Yoseph's perspective the land was indeed the "Land of the Hebrews", insofar as Hashem gave the land to Hebrews Avraham, Yitzhaq and Ya'aqov. Why would the Egyptians have been aware of Hebrews? Well, there was once an incident in Egypt with Avraham.... The Egyptians may have remembered, oh, did they remember, that Avraham the Hebrew. From their perspective Yoseph would have simply meant "the land that Avraham lived in."
In fact, the memory of Avraham may even explain why Egyptians would not dine with Hebrews: "Considering the grief that a single social contact with Abraham caused, it is not surprising that Egypt would want to prevent any further social contacts with "the Hebrew."
In any case, it is clear that the distinction between Hebrew and Jewisraelitewhatever--if there indeed originally was any--was lost on outsiders. For some reason outsiders in Tanakh regard Israelites as Hebrew.
What about the insistence of Avraham and Yitzhaq that their progeny--at least those that are Hebrew, disqualifying Yishmael--marry within, while Ya'aqov was unconcerned? Evidently this necessity of marrying within was, quite actually, a horaat shaah, a temporary measure, necessary to ensure stability in this family. Once conditions were such that it didn't matter anymore, Yehudah could marry a Canaanitess, Yoseph could marry an Egyptian etc.
As for the question of whether or not Besuel and Lavan were Hebrews--the Torah calls them Arameans--let Engelmayer take responsibility for this problem. Here are his words:
What would seem to work against this answer is Laban, who would have had to marry in for his daughters to be acceptable as wives for Jacob, yet he and his father are described as Aramaeans, not Hebrews. There is sufficient evidence, however, that both Laban and Bethuel were, in fact, believers in God, albeit not exclusively. Their daughters, however, may not have supported the apostasy.
While the text does not state this, it does offer up the possibility that this was the case. Thus, Bethuel would have been born into a home in which God was known and worshipped. Certainly, in this scenario and Laban's own statement, Bethuel's father Nahor was so raised.
Working against this is Joshua 24:2-3, which states, "Then Joshua said to all the people, 'Thus said the Lord, the God of Israel: In olden times, your forefathers--Terah, father of Abraham and father of Nahor--lived beyond the Euphrates and worshiped other gods. But I took your father Abraham from beyond the Euphrates and led him through the whole land of Canaan and multiplied his offspring.…'"
Terah, then, was an idol-worshipper, according to Joshua. This obviously conflicts with Laban's statement in Genesis 31:53 that Abraham's God was also "the God of Nahor [and] the God of their father," meaning Terah.
One way to reconcile the two is to suggest that Terah was a God-worshipper, but he had begun to slip, seeking out the support of "other gods" in addition to the God of his family. God then orders him to take his family and move to Canaan, presumably to eliminate the influence Aramaean paganism is having on them. Nahor stays behind, but Terah sets out for Canaan. The pagan influence, however, is too great on him and, before reaching his goal, Terah settles in Haran. God then transfers the call to Abraham.
To sum up, a Hebrew is the progeny of a full descendant of Eber who married another full descendant of Eber and whose house adheres to an exclusive belief in God.
As the family of Abraham grew into a nation, the need for "marrying in" lessened and, eventually, disappeared. Thus, unlike Abraham and Isaac before him, Jacob makes no effort to continue the line with his children. That was important only when the Hebrews were few and far between; Jacob's large brood meant that there were more God-fearers immediately at hand from among whom he could choose spouses for his grandchildren.
I left out portions which pertain to the definition of an eved Ivri, a Hebrew slave. According to Engelmayer at the date of the Torah's giving there were still distinctions between Ivrim and benei Yisrael, with Ivrim, perhaps, being a kind of creme de la creme of Israelite society. This explains the improved treatment that the Torah requires for an eved Ivri, who, according to this peshat wasn't merely an Israelite slave--he was an Israelite slave who was also Hebrew.