Why did God choose Israel?

What does Paul mean by "The God of this people Israel chose our fathers" (Acts 13:17)?
That God chose Abraham, his son Isaac, Isaac's son Jacob, who was later renamed "Israel," and Jacob/Israel's twelve sons to be the "fathers" (Acts 13:17, see Patriarchs) of the nation of "Israel" that God chose as "His" people.

What was so special about them that made God choose Israel from all of the people on the earth?
Abraham was a coward who instead of protecting his wife, disavowed their marriage and let two other men take her with the intent to sleep with her in the hope of saving his own neck (see Genesis 12 and Genesis 20). Like his father, Isaac volunteered the same cowardly lie concerning his wife, Rebekah: "So Isaac dwelt in Gerar. And the men of the place asked about his wife. And he said, “She is my sister”; for he was afraid to say, “She is my wife,” because he thought, “lest the men of the place kill me for Rebekah, because she is beautiful to behold.” Now it came to pass, when he had been there a long time, that Abimelech king of the Philistines looked through a window, and saw, and there was Isaac, showing endearment to Rebekah his wife. Then Abimelech called Isaac and said, “Quite obviously she is your wife; so how could you say, ‘She is my sister’?” Isaac said to him, “Because I said, ‘Lest I die on account of her.’” And Abimelech said, “What is this you have done to us? One of the people might soon have lain with your wife, and you would have brought guilt on us.” So Abimelech charged all his people, saying, “He who touches this man or his wife shall surely be put to death” (Genesis 26:6-11). Jacob was a liar, a thief and a blasphemer who lied to his father to steal from his brother and even used God's name in his lie (see Genesis 27), and Jacob's sons sold their own brother into slavery after colluding to murder him and then lied  about it to their father (see Genesis 37).

So why would God choose to mold His nation out of such a messed up family?
The same reason he chooses and molds any Christian into His image today: to demonstrate His love, grace and mercy, God chooses people who deserve absolutely none of His love, grace or mercy.

How did God exalt "the people when they dwelt as strangers in the land of Egypt" (Acts 13:17)?
God grew them into a mighty nation: "And Joseph died, all his brothers, and all that generation. But the children of Israel were fruitful and increased abundantly, multiplied and grew exceedingly mighty; and the land was filled with them. Now there arose a new king over Egypt, who did not know Joseph. And he said to his people, “Look, the people of the children of Israel are more and mightier than we" (Exodus 1:6-9, see King of Egypt).

Why does Paul say that "with an uplifted arm [God] brought them out of" (Acts 13:7) Egypt?
He is referring to God's miracles that forced Pharaoh to let the Hebrews leave Egypt, the majority of which were performed with Moses' or Aaron's arms uplifted. For example, "Then the LORD said to Moses, “Stretch out your hand toward heaven, that there may be darkness over the land of Egypt, darkness which may even be felt.” So Moses stretched out his hand toward heaven, and there was thick darkness in all the land of Egypt three days" (Exodus 10:21-22). "Then the LORD spoke to Moses, “Say to Aaron, ‘Stretch out your hand with your rod over the streams, over the rivers, and over the ponds, and cause frogs to come up on the land of Egypt.’ ” So Aaron stretched out his hand over the waters of Egypt, and the frogs came up and covered the land of Egypt" (Exodus 8:5-6). And God of course parted the Red Sea when Moses' arms were uplifted as instructed: "And when Pharaoh drew near, the children of Israel lifted their eyes, and behold, the Egyptians marched after them. So they were very afraid, and the children of Israel cried out to the LORD. Then they said to Moses, “Because there were no graves in Egypt, have you taken us away to die in the wilderness? Why have you so dealt with us, to bring us up out of Egypt? Is this not the word that we told you in Egypt, saying, ‘Let us alone that we may serve the Egyptians’? For it would have been better for us to serve the Egyptians than that we should die in the wilderness.” And Moses said to the people, “Do not be afraid. Stand still, and see the salvation of the LORD, which He will accomplish for you today. For the Egyptians whom you see today, you shall see again no more forever. The LORD will fight for you, and you shall hold your peace.” And the LORD said to Moses, “Why do you cry to Me? Tell the children of Israel to go forward. But lift up your rod, and stretch out your hand over the sea and divide it. And the children of Israel shall go on dry ground through the midst of the sea. And I indeed will harden the hearts of the Egyptians, and they shall follow them. So I will gain honor over Pharaoh and over all his army, his chariots, and his horsemen. Then the Egyptians shall know that I am the LORD, when I have gained honor for Myself over Pharaoh, his chariots, and his horsemen.” ... Then Moses stretched out his hand over the sea; and the LORD caused the sea to go back by a strong east wind all that night, and made the sea into dry land, and the waters were divided. So the children of Israel went into the midst of the sea on the dry ground, and the waters were a wall to them on their right hand and on their left." (Exodus 14:10-18, 21-22)