The words of king Lemuel, the prophecy that his mother taught him. The words of King Lemuel - דברי למואל מכך dibrey lemuel melech, "The words to Muel the king." So the Syriac; and so I think it should be read, the ל lamed being the article or preposition.
But who is Muel or Lemuel? Solomon, according to general opinion; and the mother here mentioned, Bath-sheba. I cannot receive these sayings; for
1. Whoever this was, he appears to have been the first-born of his mother: called here emphatically בר בטני bar bitni, the son of my womb; which is not likely to be true of Solomon, as his mother had been the wife of Uriah, and possibly had borne that rough and faithful soldier some children.
2. It is intimated here that this son had come by a lawful marriage: hence בר נדרי bar nedarai, the son of my vow, her matrimonial covenant; for so it is most natural to understand the words. But is there any proper sense in which we can say that this was correct in reference to David, Bath-sheba and Solomon? For although the son born in adultery died, it is by no means likely that Bath-sheba made any particular vows relative to Solomon; for of her piety, so much vaunted of by some writers, we yet want the proofs.
But, however this may be, there is no evidence whatever that Muel or Lemuel means Solomon; the chapter seems, to be much later than his time, and the several Chaldaisms which occur in the very opening of it are no mean proof of this. If Agur was not the author of it, it may be considered as another supplement to the book of Proverbs. Most certainly Solomon did not write it.
The prophecy that his mother taught him - משא massa may here signify the oracle; the subject that came by Divine inspiration; see on Proverbs 30:1 (note). From this and some other circumstances it is probable that both these Chapter s were written by the same author. Houbigant thinks that Massa here is the name of a place; and, therefore, translates, "The words of Lemuel, king of Massa, with which his mother instructed him."