“invoke Yahweh.” Although there are other words for invoking God, the Hebrew text can mean “invoke” (cp. ESV; NET; NIV). The wise woman wanted a commitment on David’s part and invoking the name of Yahweh would be such a commitment, and she pressed David until he did invoke God, using the oath formula, “As Yahweh lives.” Once she got David to make the oath, then she changed her story and it became clear she was actually talking about David’s son Absalom.
“avenger of blood.” A member of one’s family or clan that would kill anyone who killed someone in the family (see commentary on Num. 35:19).
“increase the loss.” Literally, “from the multiplying of,” which is difficult to understand and translate. The meaning seems to be that death has already robbed the woman, and she does not want the loss multiplied (cp. HCSB).
“not one hair of your son will fall to the earth.” In its historical context, this is ironic. David says of this make-believe person that not one hair of his head would fall to the earth when the real story and ruse was about Absalom, who had beautiful, abundant hair (2 Sam. 14:25-26), which got stuck in a tree and who died without a hair touching the earth (2 Sam. 18:9-17).