The ruler is establishing his covenant with his nation, as yet unborn. He reveals his name: God Shaddai. He gives his servant a new name, a new identity: he is now Abraham:
Genesis 17:3–8 (ESV)
3 Then Abram fell on his face. And God said to him, 4 “Behold, my covenant is with you, and you shall be the father of a multitude of nations. 5 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. 6 I will make you exceedingly fruitful, and I will make you into nations, and kings shall come from you. 7 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. 8 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.”
Don’t miss the significance of this as a kingdom story. It’s about nations. Kings come from Abraham. The covenant is the legally binding agreement between the heavenly sovereign and his people. Proclaiming himself God over Abraham’s descendants is not limited to the spiritual dimension: as a nation, they will have a heavenly ruler. That was Israel’s distinctiveness: while the nations suffered under human rulers, Israel’s majestic monarch was the omnipotent ruler of heaven and earth. Unlike earthly rulers, Israel’s sovereign could fulfil his promises because he reigned forever. “To be God to you” (17:7) is the promise to be their ruler. That’s the main point, so it is repeated in verse 8: “I will be their God [ruler].”
This patriarchal covenant defines the shape of the kingdom-of-God story in the Old Testament. Abraham’s descendants represent God’s kingdom among the nations.
Once you see that, you can see why Jesus expected people from other places to come and join the feast at the heavenly sovereign’s table, participating in heaven’s reign:
Matthew 8:11 (ESV)
I tell you, many will come from east and west and recline at table with Abraham, Isaac, and Jacob in the kingdom of heaven …
For this to work out, the descendants of Abraham need to keep the covenant, i.e. to serve God as their ultimate ruler, with no other rulers before him (17:9). The reality is that Israel constantly struggled with this calling, so the patriarchal covenant did not reach ultimate fulfilment until a particular offspring of Abraham restored God’s reign to his world.
But that’s the start of another book: “This is the genealogy of Jesus the Messiah the son of David, the son of Abraham” (Matthew 1:1, compare also Galatians 3 – 4).
The patriarchal covenant where God promises to be the ruler of Abraham’s descendants is a key stepping stone on the road to restoring his reign over humanity:
Revelation 21:3 (ESV)
And I heard a loud voice from the throne saying, “Behold, the dwelling place of God is with man. He will dwell with them, and they will be his people, and God himself will be with them as their God.
He will be our ruler. That’s the promise of the one who holds the throne.
What others are saying
J. Richard Middleton, A New Heaven and a New Earth: Reclaiming Biblical Eschatology (Grand Rapids, MI: Baker Academic, 2014), 89–90:
It is noteworthy that the promise “I will be their God, and they shall be my people” first shows up in connection with God’s promise of land to Abraham (Gen. 17:7–8). When YHWH later rearticulates this same promise of “relationship” to Moses in Exodus (6:7), it is sandwiched between the announcement of deliverance from bondage (6:6) and the gift of land (6:8). It thus makes sense that prophetic restatements of this ancient promise are explicitly linked with God’s dwelling with the redeemed people in a safe and bountiful land after the return from exile, since this is what is required for human flourishing (Jer. 32:37–41; Ezek. 34:25–31; 37:24–28; Zech. 8:7–8).
This trajectory of God’s presence among his people culminates in the New Testament’s vision of a redeemed creation, with the new Jerusalem coming down out of heaven to earth, and at its center is God’s throne (Rev. 21:1–22:5). Then the ancient promise will be finally fulfilled: “See, the home of God is with mortals. He will dwell with them as their God; they will be his peoples, and God himself will be with them” (Rev. 21:3).
Gordon J. Wenham, Genesis 1–15, Word Biblical Commentary (Dallas: Word, 1998), 252:
The name [Abram] is clearly composed of two elements, Ab “father” and a verb rûm which means “be high, exalted” in west-Semitic contexts, though in Akkadian the root râmu means “to love.” Assuming a west-Semitic interpretation, Ab-ram may mean “he is exalted as to his father,” i.e., he is of noble birth, or, more probably, “the father [i.e., God] is exalted.” Later Abram’s name is changed to Abraham (17:5). Probably this is just a phonetic variant of Abram, with the insertion of h into the weak root ram, a change attested in Aramaic, Ugaritic, and Phoenician. Liverani (Hen 1  9–18) has suggested that Abraham is the name of the eponymous ancestor of the rhm, a nomadic group living in northern Palestine ca. 1300 b.c. However, the popular explanation offered in Gen 17:5 understands it to mean “father of a multitude.” Though raham, “multitude,” is unattested in biblical Hebrew, one must assume that this or a very similar-sounding word was known in Hebrew for this explanation to be offered.
Read Genesis 17:4-8.