Concerning Proverbs chapter 31, who is King Lemuel and his mother?

Proverbs 31:2-9 [2] What, my son? and what, the son of my womb? and what, the son of my vows? [3] Give not thy strength unto women, nor thy ways to that which destroyeth kings. [4] It is not for kings, O Lemuel, it is not for kings to drink wine; nor for princes strong drink: [5] Lest they drink, and forget the law, and pervert the judgment of any of the afflicted. [6] Give strong drink unto him that is ready to perish, and wine unto those that be of heavy hearts. [7] Let him drink, and forget his poverty, and remember his misery no more. [8] Open thy mouth for the dumb in the cause of all such as are appointed to destruction. [9] Open thy mouth, judge righteously, and plead the cause of the poor and needy.
See All... is introduced as the words of King Lemuel from prophecy that had been taught to him by his mother. Proverbs 31:1The words of king Lemuel, the prophecy that his mother taught him.
See All... states, "The words of king Lemuel, the prophecy that his mother taught him." Lemuel is mentioned only in this passage in the Bible (Proverbs 31:1The words of king Lemuel, the prophecy that his mother taught him.
See All..., 4). This has left the door open to all kinds of speculation as to his true identity. He has been thought by interpreters to be imaginary, to be Solomon himself, to be Hezekiah, to be a Lemuel who was king of Massa (a play on the Hebrew words), or just some petty Arabian prince. In other words, no one really knows.

The name means "to God" and has the implication of "belonging to God." El (the basic name for God in Hebrew) on the end of Lemuel shows the name to be a compound of God. Personally, I think the name and context points to a poetic reference to Solomon. In Ecclesiastes, Solomon never uses his own name but presents himself seven times as the "Preacher" (Ecclesiastes 1:1The words of the Preacher, the son of David, king in Jerusalem.
See All..., 2, 12; 7:27
See All...; 12:8
See All..., 9, 10). The shift in emphasis in Proverbs would call for a different construction. Through most of Proverbs, Solomon is giving words of wisdom to his son. In Proverbs 31, King Lemuel is repeating the words of wisdom given to him by his mother.

The advice is clearly advice that Solomon needed to hear. Lemuel's mother warned her son against giving his strength unto women (Proverbs 31:3Give not thy strength unto women, nor thy ways to that which destroyeth kings.
See All...). This problem directly led to Solomon's decline in later years (1 Kings 11:1-4 [1] But king Solomon loved many strange women, together with the daughter of Pharaoh, women of the Moabites, Ammonites, Edomites, Zidonians, and Hittites: [2] Of the nations concerning which the LORD said unto the children of Israel, Ye shall not go in to them, neither shall they come in unto you: for surely they will turn away your heart after their gods: Solomon clave unto these in love. [3] And he had seven hundred wives, princesses, and three hundred concubines: and his wives turned away his heart. [4] For it came to pass, when Solomon was old, that his wives turned away his heart after other gods: and his heart was not perfect with the LORD his God, as was the heart of David his father.
See All...). She also warned against strong drink (Proverbs 31:4-7 [4] It is not for kings, O Lemuel, it is not for kings to drink wine; nor for princes strong drink: [5] Lest they drink, and forget the law, and pervert the judgment of any of the afflicted. [6] Give strong drink unto him that is ready to perish, and wine unto those that be of heavy hearts. [7] Let him drink, and forget his poverty, and remember his misery no more.
See All...). This is something we know Solomon toyed with from his testimony in Ecclesiastes 2:3I sought in mine heart to give myself unto wine, yet acquainting mine heart with wisdom; and to lay hold on folly, till I might see what was that good for the sons of men, which they should do under the heaven all the days of their life.
See All...--"I sought in mine heart to give myself unto wine, yet acquainting mine heart with wisdom; and to lay hold on folly, till I might see what was that good for the sons of men, which they should do under the heaven all the days of their life." Finally, she twice admonished her son: "Open thy mouth" (Proverbs 31:8Open thy mouth for the dumb in the cause of all such as are appointed to destruction.
See All..., 9). He is to open it in the cause who cannot speak for themselves (v.8) and to judge righteously (v.9). We know of Solomon's initial hesitancy and concern in this matter of judging the people from his own testimony in 1 Kings 3:7-9 [7] And now, O LORD my God, thou hast made thy servant king instead of David my father: and I am but a little child: I know not how to go out or come in. [8] And thy servant is in the midst of thy people which thou hast chosen, a great people, that cannot be numbered nor counted for multitude. [9] Give therefore thy servant an understanding heart to judge thy people, that I may discern between good and bad: for who is able to judge this thy so great a people?
See All.... He saw himself as a child (v.7) and desired God's help to "judge this thy so great a people" (v.9).

The words are also presented as "prophecy" given to Lemuel from his mother (Proverbs 31:2What, my son? and what, the son of my womb? and what, the son of my vows?
See All...). Prophecy does include the proclamation of God's truth, but it normally has at least an element of foretelling the future. If this refers to Bathsheba and she is telling Solomon how he will need to act when he is king, then it definitely includes a strong element of prophecy, for Solomon was a younger son and therefore not the natural one in line to be king. When David drew close to death and Adonijah set himself up as king, Bathsheba approached him with this plea: "My lord, thou swarest by the LORD thy God unto thine handmaid, saying, Assuredly Solomon thy son shall reign after me, and he shall sit upon my throne" (1 Kings 1:17And she said unto him, My lord, thou swarest by the LORD thy God unto thine handmaid, saying, Assuredly Solomon thy son shall reign after me, and he shall sit upon my throne.
See All...). David and Bathsheba had talked about it. The choice was to be Solomon. Bathsheba could prophesy the coming reign of her son as she spoke to the young prince Solomon.

One other internal evidence that Lemuel might be a poetic name for Solomon is in the mother's address to her son. Proverbs 31:2What, my son? and what, the son of my womb? and what, the son of my vows?
See All... states, "What, my son? and what, the son of my womb? and what, the son of my vows?" Lemuel is the son of her vows. The first son to Bathsheba and David was a son of broken vows. Because of those broken vows, that son had to die (2 Samuel 12:14Howbeit, because by this deed thou hast given great occasion to the enemies of the LORD to blaspheme, the child also that is born unto thee shall surely die.
See All...). However, when Solomon was born, David's sin had been revealed, confessed, and forgiven. God accepted the marriage and the son Solomon.

2 Samuel 12:24And David comforted Bathsheba his wife, and went in unto her, and lay with her: and she bare a son, and he called his name Solomon: and the LORD loved him.
See All... And David comforted Bathsheba his wife, and went in unto her, and lay with her: and she bare a son, and he called his name Solomon: and the LORD loved him. 25 And he sent by the hand of Nathan the prophet; and he called his name Jedidiah, because of the LORD.

Notice, at the birth of Solomon, the Lord loved him. Then, he is called by the "name Jedidiah, because of the LORD." This is the only occurrence of the name Jedidiah in the Bible. It means "beloved of God." Truly, Solomon was the son of Bathsheba's vows. And, the one who was "beloved of God" could also be said to be the one "belonging to God" (meaning of Lemuel). So, although the identity cannot be nailed down with absolute certainty, there is good reason to think that Solomon is here referring to himself.