The NET Bible has a useful translator's note on the introduction of the name in Exodus 3:14:
The verb form used here is אֶהְיֶה (’ehyeh), the Qal imperfect, first person common singular, of the verb הָיָה (haya, “to be”). It forms an excellent paronomasia with the name. So when God used the verb to express his name, he used this form saying, “I am.” When his people refer to him as Yahweh, which is the third person masculine singular form of the same verb, they say “he is.” Some commentators argue for a future tense translation, “I will be who I will be,” because the verb has an active quality about it, and the Israelites lived in the light of the promises for the future. They argue that “I am” would be of little help to the Israelites in bondage. But a translation of “I will be” does not effectively do much more except restrict it to the future. The idea of the verb would certainly indicate that God is not bound by time, and while he is present (“I am”) he will always be present, even in the future, and so “I am” would embrace that as well (see also Ruth 2:13; Ps 50:21; Hos 1:9). The Greek translation of the OT used a participle to capture the idea, and several times in the Gospels Jesus used the powerful “I am” with this significance (e.g., John 8:58). The point is that Yahweh is sovereignly independent of all creation and that his presence guarantees the fulfillment of the covenant (cf. Isa 41:4; 42:6, 8; 43:10-11; 44:6; 45:5-7). Others argue for a causative Hiphil translation of “I will cause to be,” but nowhere in the Bible does this verb appear in Hiphil or Piel. A good summary of the views can be found in G. H. Parke-Taylor, Yahweh, the Divine Name in the Bible. See among the many articles: B. Beitzel, “Exodus 3:14 and the Divine Name: A Case of Biblical Paronomasia,” TJ 1 (1980): 5-20; C. D. Isbell, “The Divine Name ehyeh as a Symbol of Presence in Israelite Tradition,” HAR 2 (1978): 101-18; J. G. Janzen, “What’s in a Name? Yahweh in Exodus 3 and the Wider Biblical Context,” Int 33 (1979): 227-39; J. R. Lundbom, “God’s Use of the Idem per Idem to Terminate Debate,” HTR 71 (1978): 193-201; A. R. Millard, “Yw and Yhw Names,” VT 30 (1980): 208-12; and R. Youngblood, “A New Occurrence of the Divine Name ‘I AM,’” JETS 15 (1972): 144-52.
Contrary to what you might expect from Exodus 6:3, the name appears as early as Genesis 2:4. The NET Bible notes:
Advocates of the so-called documentary hypothesis of pentateuchal authorship argue that the introduction of the name Yahweh (Lord) here indicates that a new source (designated J), a parallel account of creation, begins here. In this scheme Gen 1:1-2:3 is understood as the priestly source (designated P) of creation. Critics of this approach often respond that the names, rather than indicating separate sources, were chosen to reflect the subject matter (see U. Cassuto, The Documentary Hypothesis). Gen 1:1–2:3 is the grand prologue of the book, showing the sovereign God creating by decree. The narrative beginning in 2:4 is the account of what this God invested in his creation. Since it deals with the close, personal involvement of the covenant God, the narrative uses the covenantal name Yahweh (Lord) in combination with the name God. For a recent discussion of the documentary hypothesis from a theologically conservative perspective, see D. A. Garrett, Rethinking Genesis. For an attempt by source critics to demonstrate the legitimacy of the source critical method on the basis of ancient Near Eastern parallels, see J. H. Tigay, ed., Empirical Models for Biblical Criticism. For reaction to the source critical method by literary critics, see I. M. Kikawada and A. Quinn, Before Abraham Was; R. Alter, The Art of Biblical Narrative, 131-54; and Adele Berlin, Poetics and Interpretation of Biblical Narrative, 111-34.
I can't say anything with authority about the documentary hypothesis beyond what I just quoted. My time in the Pentateuch has convinced me that each of the books were compiled from prior sources. Genesis 1 clearly has a different textual history than most of Genesis 2, for instance. But I don't see how we can know when or who the compiler(s) were. The idea that we can identify sources from which name of God they use is clever (assuming it works). However, the idea doesn't help us date the sections without making other assumptions.
Finally the NET note on God's name in Genesis 1:1:
God. This frequently used Hebrew name for God (אֱלֹהִים,’elohim ) is a plural form. When it refers to the one true God, the singular verb is normally used, as here. The plural form indicates majesty; the name stresses God’s sovereignty and incomparability – he is the “God of gods.”
In ancient times, names held even more significance than they do now. Take Genesis 17:1-8 (ESV):
When Abram was ninety-nine years old the LORD appeared to Abram and said to him, “I am God Almighty; walk before me, and be blameless, that I may make my covenant between me and you, and may multiply you greatly.” Then Abram fell on his face. And God said to him, “Behold, my covenant is with you, and you shall be the father of a multitude of nations. No longer shall your name be called Abram, but your name shall be Abraham, for I have made you the father of a multitude of nations. I will make you exceedingly fruitful, and I will make you into nations, and kings shall come from you. And I will establish my covenant between me and you and your offspring after you throughout their generations for an everlasting covenant, to be God to you and to your offspring after you. And I will give to you and to your offspring after you the land of your sojournings, all the land of Canaan, for an everlasting possession, and I will be their God.”
According to the ESV notes, Abram means "exalted father" and Abraham means "father of a multitude". Now Abram was a father at 86 of Ishmael by Hagar (Genesis 16) and so his name had finally been fulfilled after a fashion. When God changed his name to Abraham, it had significance both because he received a new promise and the covenant changed his relationship to God. It was a sort of sign and a way to remember the promise.
Note that ’elohim ("God"), Yahweh ("LORD") and another name, El Shaddai ("God Almighty") are used in this short passage. Directly contradicting Exodus 6:3, Abram did invoke the name of Yahweh in Genesis 14:22-23 (ESV):
But Abram said to the king of Sodom, “I have lifted my hand to the LORD, God Most High, Possessor of heaven and earth, that I would not take a thread or a sandal strap or anything that is yours, lest you should say, ‘I have made Abram rich.’
I believe we can resolve that contradiction if we go back and look at the context of Exodus 6:2-8 (ESV):
God spoke to Moses and said to him, “I am the LORD. I appeared to Abraham, to Isaac, and to Jacob, as God Almighty, but by my name the LORD I did not make myself known to them. I also established my covenant with them to give them the land of Canaan, the land in which they lived as sojourners. Moreover, I have heard the groaning of the people of Israel whom the Egyptians hold as slaves, and I have remembered my covenant. Say therefore to the people of Israel, ‘I am the LORD, and I will bring you out from under the burdens of the Egyptians, and I will deliver you from slavery to them, and I will redeem you with an outstretched arm and with great acts of judgment. I will take you to be my people, and I will be your God, and you shall know that I am the LORD your God, who has brought you out from under the burdens of the Egyptians. I will bring you into the land that I swore to give to Abraham, to Isaac, and to Jacob. I will give it to you for a possession. I am the LORD.’”
So the LORD recalls the promise he'd made to the patriarchs that seemingly has not and will not be fulfilled. Israel is stuck in Egypt as slaves. Genesis makes clear that the patriarchs never laid claim to any land in Canaan except a burial plot. Pagan gods are fickle and easily change their minds. But here, God tells Moses and Israel that He Is and Was and Will Be. Ironically, God suggests that He is unchanging by changing His name to be the unchanging "I AM". Only it isn't really God that's changing, but the relationship between God and His people. God is about to embark on a 40-year process of unparalleled miracles to bring Israel into the land He promised Abraham. Whenever the sons of Abraham think of the LORD, they recall the Exodus.
In this passage, God is commemorating the beginning of a new relationship with Israel by giving them a new name, recalling His eternal existence, by which to call Him.