Sean M. answered 08/28/19
Oxford-style Writing Tutorials with an Award-Winning Former Prof
You're on the right track when you mention Greek mythology. The River Styx is the river that souls must cross to enter the underworld. It marks the transition from the living to the dead. In that sense, it represents the giving up of all of a living person's hopes and dreams.
Early in Dante's Inferno these famous lines appear: "Lasciate ogne speranza, voi ch'intrate" or, in English, "Abandon all hope, you who enter here."
So Riordan's use of that epithet isn't all that original. It's drawing on a well-known literary reference.