Why did her son fall ill?
Rashi (from Sanhedrin 113a): This was so Eliyahu would need the key of Techiyas ha'Mesim to revive him [and Hashem would reply, I have three keys that I never entrusted to a Shali'ach 1 - birth, rain and Techiyas ha'Mesim. It is not proper that the Talmid hold two, and the Rebbi holds only one! Return the key of rain to Me, and I will give to you the key of Techiyas ha'Mesim.]
Rashi (113b): I never entrusted them to a Shali'ach other than you. Tosfos (Ta'anis 2a) - I never entrusted them to a Shali'ach permanently.
Why is she called "Ba'alas ha'Bayis"?
Malbim: She was primary in the house. The miracle was due to her.
What is the meaning of "a Neshamah did not remain in him"?
Radak #1: Some say that he did not die 1 , but he was so weak that his Neshimah (breathing) stopped, and no sign of life was left in him, i.e. breathing or pulse. His mother thought that he died. It says similarly about Daniel (10:17) "u'Neshamah Lo Nish'arah Vi."
Radak #2: It is proper to say that he truly died, like [virtually] everyone says.
Kli Yakar: This is wrong. Refer to 17:20:1:2*, Melachim II, 4:31:2:2 and the note there.
