In Divrei ha'Yamim (1:3:1), it says that his second son was Daniel!
Rashi and Radak citing Tanchuma (Toldos 6): Scoffers of the generation said that Avigayil was already pregnant from Naval (with Daniel) when David married her. 1 Therefore, Hashem made Daniel Kulo Av 2 .
Rashi citing Brachos 4a: He is called Kil'av because he was HiChLim (shamed) Mefiboshes in Halachah -- "Chacham Beni v'Samach Libi v'Ashivah Chorfi Davar."
Radak: He had two names. There are many people like this.
Malbim: This is difficult. Surely David waited the three months (from when Naval died) before marrying her, to avoid all doubts about lineage of her children! (Perhaps this was not yet enacted. Perhaps David relied on the majority of births, which are in the ninth month of pregnancy, so it suffices to wait one month! - PF)
Radak: I.e. totally like his father's form, to remove suspicion. He called him Daniel, for Danani Kel mi'Naval (Hashem judged between us, and vindicated me - PF).
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).
Why is she called Eshes Naval here, and in Divrei ha'Yamim (1:3:1), Avigayil ha'Karmelis?
Malbim: Here she is called Eshes Naval, for it says here that her son was called Kil'av, because she was previously married to Naval (refer to 3:3:1:1).
