The "Aetos Dios" (sometimes Aetos Dias) [Greek. αετός Δίας], is translated from Greek into "Eagle of Zeus". There are two schools of thought regarding the origin of this eagle, coming from different Greek legends.
Periphas was one of Apollo's priests. He was later made King on account of his many virtues. Zeus however, was jealous because King Periphas was now revered and honoured to the same extent that he was and so wished to destroy him. Apollo intervened however, and requested that Zeus instead transform the King into an eagle. Zeus accepted and transformed both King Pereiphas and his wife Phene into an eagle and an osprey respectively. The eagle also has a place amongst the stars as the constellation Aquilla, one Ptolemy's 48 listed constellations.
(Some accounts say Phene was turned into a vulture, not an osprey, and is the reason for the constellation of Lyra (also on Ptolemy's list). The constellation of Lyra is an instrument of Apollo, but is also sometimes depicted as a vulture. This could be because Lyra's brightest star, Vega, has Arabic roots meaning "Swooping Eagle". The Latin name for this was "vultur cadens", or "sinking vulture". The addition of the bird so close to the constellation of Aquilla, has led some to believe that Lyra was in fact the consort of Periphas in his eagle-state[1])
From then on the eagle would have the privilege of being able to approach the throne of Zeus. He would serve with Zeus and be the protector of Zeus' sacred sceptre, while his wife would have the privilege of being a sign of good omens in the affairs of men.[2][3]
In other accounts the eagle was in fact an ancient creation of the goddess Gaia. He appeared before Zeus at the start of the Titanomachy (Battle of the Titans). Zeus took this to mean a good omen of victory, leading to him using the emblem of a golden eagle on his war standard:
"...For so happy an omen, especially since victory did ensue, he made a golden eagle for his war standards and consecrated it to the might of his protection, whereby also among the Romans, standards of this kind are carried." - a translated excerpt from Fulgentius' "Mytholgies" (Mythologiarum Libri III)