Why should Yonason not die?
Malbim #1: (a) Even if he sinned, he did a great salvation. The nation will risk their lives for him, just like he risked his life for them, and protect him. (b) Heaven forbid to say that he sinned b'Mezid. Hashgachah clung to him today! He did not eat bread; he only tasted Devash. Hashem is meticulous with Tzadikim to consider Shogeg like Mezid, but he is not Chayav Misah as if he was Mezid!
Malbim #2: This is a reason to regret and permit his vow (lest Yonason, who did this great salvation, die through it). 1
This requires investigation why this is not Nolad (regret due to later events that one could not have anticipated). We do not permit based on regret due to Nolad (YD 228:12).
What is the meaning of "Im Elokim Asah ha'Yom ha'Zeh"?
Radak: Yonason did good with Hashem and His nation. He raised his Bitachon and risked his life for Am Hashem.
Radak citing Targum Yonasan: It is revealed in front of Hashem that he did (ate) b'Shogeg today. 1
Radak: If the exemption is due to Shegagah, why does it say "who did this great salvation"? Because he did this great salvation, it would be evil and bitter for us if we did not find this reason to exempt him. Also my father explained so.
How did the people redeem Yonason?
Rashi: They let (Malbim - insisted that) Sha'ul permit his Shevu'ah.
Radak citing R. Elazar in Midrash Shmuel 17:3: They gave his weight in gold, and redeemed him.
Radak citing R. Yochanan in Midrash Shmuel 17:3: Did he eat bread?! He ate honey! (He did not transgress the Shevu'ah.)
Radak citing Reish Lakish in Midrash Shmuel 17:3: He only tasted! Tasting is not called eating or drinking. It does not cancel a fast, and it does not require a Brachah.
Ramban (Vayikra 27:29): It was not really redemption - "Kol Cherem? Lo Yipadeh 1 " (27:29)! Rather, the salvation revealed that he was Shogeg, like Targum Yonasan says.
R. Elazar (refer to 14:45:3:2) holds that they redeemed him with gold! Sometimes when Pidyon is not needed, Chachamim obligated to give money, lest people say that that Hekdesh becomes Chulin for free. However, here we should be concerned lest people say that Pidyon can exempt from Misah! Perhaps we are not concerned, for only a Sanhedrin can judge capital cases; they would not err about this. (PF)
Targum Yonasan says that they exempted Yonason because he was Shogeg. If so, why did the Urim v'Tumim not answer?
Radak citing Rav Sadya Gaon: It was to show the nation that he was exempted because he was Shogeg. They were saying that it is due to favoritism, for he is the king's son; had someone else sinned, he would have been punished. They did not know that Yonason was not there at the time of the Shevu'ah. 1 The Urim v'Tumim did not answer, they cast lots, investigated, and found that he was not there at the time of the Shevu'ah.
Malbim citing Mahari: Sha'ul was not answered due to his own sin in Gilgal (offering a Korban before Shmuel came). 2
Malbim: Hashem is meticulous with Tzadikim to consider Shogeg like Mezid. After Yonason heard that he transgressed the Cherem, he should have tried to coax his father to permit his vow and nullify it retroactively. Since "Achar Avi Es ha'Aretz" (29), and the vow did not help (rather, it harmed), he would regret it from the beginning.
Verse 17 says that that they checked, and found that Yonason and his youth were not there [when they saw the Pelishtim in anarchy]. However, this does not show that they were not there at the time of the Shevu'ah. (PF)