How did David select which will be killed?
Rashi, citing Yevamos 79a: He passed them in front of the Aron. Whomever the Aron absorbed would be killed; whomever the Aron expelled would live.
How did David have mercy on Mefivoshes?
Rashi, citing Yevamos 79a: He prayed that the Aron not absorb him 1 (refer to 21:7: 1:1).
Radak (1) citing Rav Sadya Gaon: Mefivoshes had not killed any Giv'onim; he was only five years old when Sha'ul died (4:4)! The Giv'onim did not ask to kill him! "Va'Yachmol" means that he rejoiced [that he would not be killed]. This is like "Ka'asher Yachmol Ish Al Bno ha'Oved Oso" (Mal'achi 3:17). A son who serves does not need mercy; his father rejoices over a son who serves him. David was happy that he did not need to transgress the Bris and Shevu'ah between himself and Yehonason.
Radak (1): (a) The Giv'onim asked to kill only seven, and there were seven without Mefivoshes, therefore he mercy on him due to had the Bris and Shevu'ah between himself and Yehonason, and did not give him to the Giv'onim. (b) It seems that Mefivoshes was better than the others, and it was not proper that the punishment for Sha'ul should fall on him. This was Tikun ha'Olam, to instill fear on people not to harm converts.
Malbim: David was able to save him, because Mefivoshes' father did not agree to destroy Nov, for he had a Bris with David. 2
Yevamos 79a: You cannot say that he did not pass him in front of the Aron, or that after the Aron absorbed him. David requested mercy, and the Aron expelled him. That is favoritism!
Perhaps initially the Giv'onim wanted to kill all of Sha'ul's remaining descendants; David was able to persuade them to exempt Mefivoshes, therefore they requested only seven. Alternatively, after they requested seven, David was able to save him by persuading the Giv'onim that they have no complaint against Mefivoshes. (PF)
