Just a few thoughts taken from various essays I've been reading:
1. The tribe of Levi was set apart as a tribe of priests, mediating between God and his people, the Israelites.
2. The nation of Israel was set apart as a "kingdom of priests" (Exodus 19:6), called to be a conduit through which God blessed all the nations (Genesis 12:2-3), and through which God's name is made known (Psalm 67).
3. After Christ, we are a "priesthood of believers" meant to proclaim God's excellence to the entire world (1 Peter 2:9).
4. Christ is the ultimate High Priest, interceding for us (Hebrews 5:1-10).
When all of this is put together, it seems less strange that God would, in the middle of Genesis, set apart a seemingly random group of people to be His. This setting apart isn't an act of exclusivity. God meant to use them to bless the nations just as God used the tribe of Levi to bless His people. And when Christ died and the veil separating the Holy of Holies was torn, the separation between Jews (the kingdom of priests) and the Gentiles was removed, just as the separation between the Levites and the rest of the Jews was removed (the Levites were no longer needed). Our access to God is now through the ultimate High Priest, Jesus. And all believers (Jews and Gentiles) are called into a royal priesthood so that we might bring God to those who have yet to believe.
tl;dr - When God set apart the Jews, He meant to use them to bring the other nations to Him. God sets us apart so that He can use us to bring every nation to Him.