The play's original Greek title is Prometheus Desmotes: "Prometheus in Chains," or "Prometheus Bound." So, chains, like fire, are there from the very beginning. But "chains" are just a symbol for confinement—and for freedom, beginning when Hephaestus announces that he has come to "nail you [Prometheus], with metal bonds hard to undo" (19-20).
The motif of chains continues in the climactic dialogue where Prometheus tells Hermes, "I wouldn't exchange my misfortunes for your servitude" (964-965). And that brings us to the major questions that the symbol of chains raise: what is freedom? And what's confinement?
Prometheus may be chained, but he's really the character who shows the most freedom: he does what his conscience tells him to. In contrast, Hermes may have cute little wings on his sandals, but he's nothing more than a servant.