Eluréd and Elurín were both named after their great-grandfather Elu Thingol, King of Doriath. Their names are Sindarin for "Heir of Elu" and "Remembrance of Elu", respectively. They had a sister Elwing.
The infants Eluréd and Elurín lived with their father in Menegroth, but during the Sack of Menegroth by the Sons of Fëanor they were seized by the servants of Celegorm and abandoned in the forest to die. The eldest son of Fëanor, Maedhros, repented of the deed and sought in vain for them.
According to The Silmarillion the children were never found, and were presumed to have perished in the forest. Tradition among the Nandor of Ossiriand held that they were led to the safety of the woodlands by birds and beasts, and survived.
^HoME XI.350. “The cruel servants of Celegorn seize Dior’s sons (Elrún and Eldún) and leave them to starve in the forest. (Nothing certain is known of their fate, but some say that the birds succored them, and led them to Ossir.)”