Argon was the fourth child of Fingolfin, High King of the Ñoldor in Beleriand.
His mother was Anairë, and his brothers were Fingon, and Turgon. He had an older sister, Aredhel.
His Quenya name was Arakáno ("High Commander"), named so partially for his character. Argon set foot on Middle-earth with his father and brothers, but he was killed during the Battle of Lhammoth.
Arakáno thus never received a true Sindarin name, but Argon was later devised for him in the records. It is recorded that this later became a popular name under the Eldar.
Other versions of the legendarium
Argon does not appear in the published The Silmarillion at all, as he only came to light in very late writings by Tolkien.