Argus-eyed means ‘vigilant’ or ‘extremely observant’.
It refers to Argus (or Argos), a Greek mythological giant with a hundred eyes. He is usually pictured as a man with eyes all over his body.
There are slightly different versions of the story but the following is the basics of it.
Argus served the goddess Hera, and he was famous for slaying the monster Echidna (she was half-woman, half-snake).
Hera’s husband Zeus had been ‘romantically pursuing’ a nymph named Io. When Hera nearly discovered them together, Zeus transformed Io into a white cow. He claimed it was just a cow – definitely not a woman he had pestered into having sex with him. Hera was not fooled and she asked to keep the cow. Zeus had no choice but to agree.
Argus was tasked with guarding the cow. To help him, Hera gave Argus the gift (or curse) of sleeplessness. His eyes would never tire.
Hera honoured Argus by placing his eyes on the tail of her favourite bird, the peacock.