The opening verse reads: "The words of king Lemuel. The vision wherewith his mother instructed him." The name occurs again in verse 4: "Give not to kings, O Lemuel, give not wine to kings ..." The discourse, which is an exhortation to chastity and temperance, appears to end with verse 9, but might continue through the end of the chapter.(which is the end of proverbs)
Some modern scholars (see Revised Version, Prov. 31:1, margin) render the first passage thus: "The words of Lemuel, king of Massa, which his mother taught him." Massa is mentioned in Genesis 25:14 (cf. 1 Chronicles 1:30), among the sons of Ishmael, and his kingdom is consequently supposed to have been in Arabia.
Some  suspect that Lemuel is actually Solomon who is credited with most of the rest of proverbs. They find that the advice given is especially appropriate to Solomon, and that the story of Solomon's birth lines up well with the title "son of my vows" in verse 2
The name is related to Lael found in Numbers 3:24 meaning a man consecrated "to God". In etymological form the name Lemuel is kindred with Jamuel (Genesis 46:10) and Namuel (1 Chronicles 4:24).