Eridanus was one of the original 48 constellations first drawn by Ptolemy, but its date of actual origin is unknown. It has been thought to be the river of tears wept by the Heliades at the death of their brother Phaethon. Phaethon was hurled by Jove into the river after he attempted to drive the chariot of the sun.
However, Eridanus is a southern constellation and doesn't fit well with this particular myth. Some think it refers to the Nile; such references as Eratoshenes, Germanicus and Hyginus all refer to Nilus in this area of the sky.
The Euphrates river may be a better fit; both rivers are similiar, in that they wind and both have two major branches in them, and both have references to connecting heaven and hell. In ancient times, the Nile was known as Ioma, or Iauma, the Sea, whereas the Euphrates was known as Pura, or Purat, the River. To the Arabs, however, the constellation was associated with Nar, or Nahar, which translated to the Nile.