For what purpose were the "Chosen People" chosen?

There's a story from the Oral Torah which I believe adds an interesting element to this idea of God having "chosen" or "favored" one particular people.

http://judaism.about.com/library/3_askrabbi_o/bl_simmons_abrahamidols.htm

... when Abraham was still a young child, he realized that idol worship was nothing but foolishness. To make his point, one day, when Abraham was asked to watch the store, he took a hammer and smashed all the idols - except for the largest. His father came home aghast. "What happened?!" he shouted. "It was amazing, Dad," replied Abraham. "The idols all got into a fight and the biggest idol won!"

The idea, of course, was to show his father how ridiculous it is to ascribe power to such idols!There was no way for his father to respond; deep down he knew that Abraham had tuned into a deeper truth.

Here, in a sense, we have Abraham "choosing" God. Or, if you will, cooperating with God's Will rather than the more common idea of God choosing/electing/predestining Abraham. It's a remarkable parallel to the synergism/monergism debate. IMO, the Covenant makes more sense in this light than in the arbitrary selection by God of one particular tribe.

As the other answers suggest, there are lots of reasons why God has a chosen people.

Just want to add few points.
Lets go back to the creation where God decided or chose to create heaven an earth. He also chose to create humanity to populate the earth. And then begins a privileged relationship between God and his creation. In this relation was a covenant between God and Man. But this relationship is broken by sin and so is the covenant. As time goes, the population on earth increases along with deviant behaviors.

I believe God didn't want His first intention in creation to be lost. So He chose a man with whom He would have a covenant as Genesis 12 says:

The Lord had said to Abram, “Leave your country, your people and your father’s household and go to the land I will show you”

“I will make you into a great nation and I will bless you; I will make your name great, and you will be a blessing.

I will bless those who bless you, and whoever curses you I will curse; and all peoples on earth will be blessed through you.”

And through that chosen person, God would reach out to the entire humanity.
Jesus is the finalization of God's purpose, which is to have a new covenant with not only His chosen people (Israel) but to all humanity, which is resumed by John 3:16 - 17

For God so loved the world that He gave His only begotten son, so that whoever believes in Him, should not perish but have everlasting life. For God did not send his Son into the world to condemn the world, but to save the world through him

So through Abraham, God intended to restore His covenant with His creation.

The term chosen can also be interpreted as favored. In the most general sense, the seed of Abraham is chosen in the sense that Christ would come through Abraham's lineage.

Also, the Priesthood, or the authority to act in God's name on earth, would be passed down through the descendents of Abraham:

Abraham 2:9

And I will make of thee a great nation, and I will bless thee above measure, and make thy name great among all nations, and thou shalt be a blessing unto thy seed after thee, that in their hands they shall bear this ministry and Priesthood unto all nations;

Any follower of Christ (regardless of lineage) is considered Abraham's seed by adoption, which allows those who are not direct descendants of Abraham to hold the Priesthood.

Galations 3:29

And if ye be Christ’s, then are ye Abraham’s seed, and heirs according to the promise.