Elmo is only mentioned in the Unfinished Tales in discussions on Galadriel and Celeborn. Because Celeborn is in the Silmarillion presented as a "kinsman of Thingol", Tolkien wished to develop this family link. One of his solutions was to establish Celeborn as the grandson of Elmo, a non-important brother of Elwë who remained behind when Elwë was lost, becoming one of the Sindar of Doriath. In this conception Elmo has a son named Galadhon, who in turn sired Celeborn and another son named Galathil, who was the father of Queen Nimloth of Doriath.
It therefore remains uncertain if Elmo would have remained in the Middle-earth legendarium.