If the present ordering of the text is correct, this constitutes a expansion on the part of the woman, who has already been told that the king would take care of the matter. This may affect the translation in some languages.
Pray: see the comments on 13.6 regarding this means of introducing a request. The same particle that is found here is the one left untranslated by Revised Standard Version in verse 2 above. It is also used again in the following verse. New Revised Standard Version renders it “Please” in both cases.
Let the king invoke the LORD: since the woman is speaking directly to the king, it will be more natural in most cases to say something like “Please [you] pray…” or something similar. However, the idea contained in the verb invoke (literally “remember”) is understood by some to refer to a more formal swearing instead of praying. New Jerusalem Bible, for example, says “pronounce the name of Yahweh…,” and Moffatt says even more directly “let the king swear by the Eternal….”
The avenger of blood: this expression refers to a person who is closely related to someone who has been harmed or killed. This relative was responsible for punishing the person who had done wrong. Early in their history the people of Israel felt that this was a means of bringing about divine justice. See Num 35.19-28; Deut 19.6; and Josh 20.3-5. Revised English Bible translates “the next-of-kin.” And New Century Version attempts to make the meaning quite clear, with “my relative who has the duty of punishing a murderer.” But this sounds too general because of the indefinite article, “a murderer.” The woman is talking about the specific murder of her son and not just any murder.
As the LORD lives: see 4.9 and 12.5. Compare also 1 Sam 14.39 and Ruth 3.13. David is now taking the oath as the woman requested.
Not one hair of your son shall fall to the ground: the same idiom in a slightly different form is found in 1 Sam 14.45. This promise should probably not be translated literally in most languages. The commitment of the king is that no harm whatsoever will come to the woman’s son. There is a wide variety of ways in which such an idea may be expressed in different languages. The English idiom “no one will touch even a hair on his head” (Contemporary English Version) retains something of the biblical image, but this is not essential to the meaning of the expression. Some may say “no one will touch him,” or “people will not dare even to speak against him,” or “he will not be harmed at all.”
Quoted with permission from Omanson, Roger L. and Ellington, John E. A Handbook on the First and Second Books of Samuel, Volume 2. (UBS Helps for Translators). New York: UBS, 2001. For this and other handbooks for translators see here .