Genesis 20:12-15 meaning

Abraham discloses to Abimelech that is statement about Sarah was not untrue. She was the daughter of [Abraham's] father through their common father, Terah. However, Sarah was not the daughter of my mother, which made her eligible to be Abraham's wife. 

Abraham tells Abimelech of his request of Sarah when God caused him to wander from his father's house. Abraham asked her as follows: "Everywhere we go, say of me "he is my brother." This let Abimelech know that Abraham did was not picking on him. This had been their practice all along, during their travels. We already saw this in Egypt, where a mirror episode took place (Genesis 12:10-20). 

As we saw in Genesis, Abraham's calculation was that local leaders would kill Abraham in order to make Sarah eligible for them to marry. Sarah's courage is evident. She willingly accompanied Abraham on this journey, knowing they were entering hostile lands. She showed great bravery. This is likely one of a number of reasons she is exalted as a great example in the Bible (1 Peter 3: 1-6; Hebrews 11:11). 

It is easy to second guess Abraham from the safety of our couch. However, the text does not criticize him. God could have chastised him after the similar episode in Egypt, but does not. In each case, God's action is to protect them both.

Is Abraham inappropriately testing God? The Bible does not indicate that is the case. It could be that Abraham's calculation was 1) the statement that Sarah was his sister was true (as he argues to Abimelech in this passage) and 2) his choice was between being killed without having a chance to save Sarah, or buying time to find a way to redeem her. Current custom would require the local leader to negotiate with him, as Sarah's brother, to arrange the marriage. We have already seen Abraham's willingness to go to war in order to redeem his nephew Lot (Genesis 14: 15-16). In that episode, Abraham was very strategic, attacking by night. 

Perhaps he wanted to make sure that if he was not able to extract Sarah without a fight, he would make sure he fought on his own terms. Also, when Abraham rescued Lot, he fought invaders, not his neighbors. It seems he was going to great lengths to maintain good relationships. It is arguable that Abraham would be testing God if he did not use his own wits and capabilities to the best of his ability to protect his wife, while living at peace with his neighbors if possible, and God honors him in that pursuit. At the end of the episode, Abraham's goal seems to be granted, as he is granted to live in peace with his neighbors, as Abimelech says to him "Behold, my land is before you; settle wherever you please" 

It could be that Sarah's willingness to protect Abraham from being slain (by agreeing to say she was his sister) is in view when Sarah is commended in 1 Peter 3:6. Here it is stated that Sarah is an example of a godly wife that honors her husband, "calling him lord." The claim 1 Peter 3 makes is that when women honor their husbands in this manner, that even when the husbands are not following the word of God as they ought, that often they will be won over by the great character and responsiveness of their wives. Since believers are collectively called the bride of Christ, this example can be applied to believers living in a fallen world, being willing to follow Christ without fear in a hostile world.

 The practice of marrying a half sister would later be forbidden (Leviticus 18:9, 11, 20:17; Deuteronomy 27:22). To restart the human race after the flood marrying close family was a necessity. Abraham is only ten generations removed from Noah (Genesis 11:10-26).  It seems likely that God disallowed the practice of marrying half sisters as the genetic pool began to degrade. 

 The king then took sheep and oxen and male and female servants and gave them to Abraham, and restored his wife Sarah to him. We are not told what would have happened without God appearing to Abimelech in the dream. However, both Abimelech and Pharaoh responded favorably to God's intervention, and restored Sarah to Abraham. This in contrast to a subsequent Pharaoh who will refuse to bow to God's demands to let His people go, in spite of supernatural intervention (Exodus 6-15). 
Abimelech invites Abraham to settle wherever he pleases in the land of Gerar. He would no longer be considered an alien in Gerar. In contrast, the Pharaoh had his men escort Abraham and his household out of Egypt (Genesis 12:20). God instructed Abimelech that Abraham would pray for you and you will live. It seems that Abimelech reasoned from that instruction that his best course of action was to stay completely on Abraham's good side.