The words, &c. The Hebrew dibrey has no governing noun; so one must be understood thus. "[An unnamed one's] words [addressed] to king Lemuel: a burden which his mother [repeatedly] taught him".
king Lemuel = to El and for El, a king. Compare Jedidiah = beloved of Jah (2Sa_12:25). Solomon was the royal seed in the line of Him Who is King of kings and Lord of lords. The Talmud says (Avoth d'Rab. Nathan, c. 39): "Solomon was called by six names: Solomon, Jedidiah, Koheleth, Son of Jakeh, Agur, and Lemuel".