Why did she say that he came to mention her sin?
Rashi (from Bereishis Rabah 50:11): Before you came to me, Hashem compared my deeds to the deeds of [others in] my city, and I was worthy of a miracle. From when you came to here, I am not considered anything (special), and my virtue is not recognized. Lot similarly said (Bereishis 19:19) "I cannot flee to the mountain, by Avraham", for my merit will not be recognized when I am by him.
Pirkei d'R. Eliezer (33): People will say that "Basa Elai" (you had Bi'ah with me), and [via this] you mentioned my sin, and my son died.
Radak: It seems that via you, my sin was mentioned. Perhaps I did not serve you properly, and this mentioned my sin, and Hashem punished me through the death of my son.
Malbim (17): It was recognized that his illness was a punishment. (a) Hashem would do a miracle to save her son [from starving], and [right] afterwards he will die. 'The One who gives food, He gives for life!' (Ta'anis 8b) (b) Normally, an illness begins mild and gets more severe. Here, he immediately became mortally ill.
Malbim: (a) She suspected that his words that Hashem's word brought the Brachah were not true. Rather, he did the miracle via his power, names of Hashem or Segulos to change nature, and therefore her son died, for benefiting from witchcraft or supernatural matters. This is considered a sin, and he was Chayav Misah - 'one who uses the crown [of Torah] is Chayav Misah 1 ' (Avos 1:13). (b) If a miracle is done for someone, they deduct from his merits (Ta'anis 20b). If reality must be changed for someone, they weigh his merits and sins, to see if he is worthy of this. This would not apply if the miracle was based on Hashem's will, without looking at merits and guilt.
If so, why did only her son die, and not her, and not Eliyahu, who used these powers b'Mezid? (PF)