Genèse 20:12

But in truth she is my sister, the daughter of my father's brother, but not of the kindred of my mother; and she became my wife.

AND YET SHE IS MY SISTER, THE DAUGHTER OF MY FATHER. I know not the sense of this apology. Even if it were true that she was his sister and his wife, nevertheless when they wanted to take her as a wife and he told them, She is my sister, in order to lead them astray, he already committed a sin towards them by bringing upon them a great sin, and it no longer mattered at all whether the thing was true or false!
Perhaps it was because Abimelech said, What sawest thou, that thou hast done this thing? meaning “What sin or wickedness have you seen in me that caused you to do this out of fear? I have never attempted to take women away from their husbands.” Then Abraham answered: “I did not know you, but I thought, perhaps the fear of G-d is not in this place, for in most places in the world there is no fear of G-d. Therefore, from the time I left my country and aimlessly wandered among the nations not knowing to which place we would come, I made this condition with her that she say thus in all places. For the matter is true, and I thought that by doing this, human life would be saved. I did not begin this practice upon entering your country for I did not see that you had committed any sin.” And yet she is my sister, the daughter of my father. This expression represents a different argument. Abraham said: “According to my custom, I said so for it is true, and I further thought that in case they will want her they will ask me if she is also my wife. Since your servants took her and they did not ask me any questions, I said, ‘The fear of G-d is also not in this place’ and I remained quiet.”
It is possible that he established this condition with her when they came to Egypt even though he said that he did it when G-d caused me to wander from my father’s house. It may be that he again warned her there in Egypt at the time of the event, as I have explained.
Rabbi Abraham ibn Ezra’s opinion is that all these words were to put off Abimelech.

אחותי בת אבי, grandchildren are treated just like children and Sarah was a granddaughter of Terach.

אך לא בת אמי, however she is not a sister who was born to my biological mother. This is why she is permitted for me to marry, and she became my wife.

וגם אמנה, also you, the king himself, have sinned in taking the woman because I said she is my sister for you should have asked me if she was also my wife. The fact is that she is both my sister and my wife.

וגם אמנה, besides she is indeed my sister. The expression אמנה is equivalent to באמת, “in truth.” The word occurs in the same sense in Joshua 7,20 אמנה חטאתי, where Achan admits having stolen from the booty of Jericho. The word אמנם or אמנה both mean basically the same. Avraham in effect told Avimelech: “even though I did not protest the violence committed against Sarah this does not make me a liar, as she is indeed my sister, being the daughter of my brother, Legally, nieces are just as children as grandchildren are considered as legally equivalent to children.” (compare Rashi.

אך לא בת אמי, seeing Avraham’s brother Haran had been born to his father Terach by another wife than his mother. Furthermore, seeing that Avimelech was certainly unaware of this, it would have been quite in order for Avraham to marry Sarah even if she had been the granddaughter of Terach from the same wife as the one who had born Avraham. Our sages, taking their cue from what Avraham said here, state that a paternal sister is not of the same category as a maternal sister, only the latter being considered as full a sister. (Sanhedrin 58). According to this, Avraham would have been entitled to marry a first generation paternal sister, not only a second generation paternal sister such as Sarah. He would not have been allowed to marry a first generation maternal sister, however.