Why does it say "l'Avshalom"? This implies his children!
Radak: This is like Es Avshalom. Also "Hargu l'Avner" (Shmuel II, 3:30), "l'Chol Kelav Ta'aseh Nechoshes" (Shemos 27:3) and others (the prefix Lamed is like "Es").
Vilna Gaon: He was from a Yefas To'ar, and was killed for rebellion against his father. He was not worthy that his lineage be written. It was written only for the sake of the lineage of his children.
Malbim: Because his name is like Avi ha'Shalom (and just the contrary, he rebelled!), Lamed was added to change his name for detriment, like Lo Av Shalom.
Why does it say "Ben Ma'achah" and "Ven Chagis"? For the other sons, it says [that they were born] 'la'Plonis'!
Vilna Gaon: Everywhere Adoniyahu is attributed to his mother and called Ben Chagis. (The Vilna Gaon says similarly about Avshalom; refer to 3:2:1:2.)
Malbim: His mother was a Yefas To'ar. This caused him to rebel against his father. Also Adoniyahu is called Ben Chagis, because she raised him to be evil (Sanhedrin 107a). "His mother gave birth to him after Avshalom" (Melachim I, 1:6) teaches that she raised him to go on the same [evil] path that Avshalom's mother raised him (Rashi there).
What is the significance that Avshalom's mother was the daughter of Melech Geshur?
Tanchuma (Toldos 6): She was a Yefas To'ar that David took in war. The Torah wrote Ben Sorer u'Moreh next to Yefas To'ar, to teach that one who marries a woman not proper for him, he will have a rebellious son. Radak - this Drashah is proper. Since she is forced to convert, she does not truly intend for Torah; the tree (child) is like the root (mother).
