The role of Hades is one of James Woods's all-time favorite characters. So much so, any time Disney needs him to reprise the character for any cartoon show and video game, such as Hercules (1998) or Kingdom Hearts, he has agreed to do so. When this film was about to go over-budget, he offered to refund his salary and finish the film for free. Disney decided against this, and finished the film anyway.
Hades was scripted as a slow, menacing character until James Woods took the role and impressed the crew with his rapid-fire style. Even after the script was rewritten, Woods ad-libbed many lines.
Hades says "Relax, it's only half-time" right in the middle of the movie.
In the movie, Pain and Panic take the shape of two snakes in order to kill infant Hercules, but he simply grabs and tosses them about. This is another reference to the Herculean myths, where goddess Hera sent two snakes to kill infant Hercules, who simply grabbed and choked them.
When Phil and Hercules first enter Phil's hut, Hercules bangs his head on what Phil tells him is the mast of the Argo. In Greek mythology, Jason, the captain of the Argo, was killed when the mast hit his head. In the myths, Hercules was one of the heroes/sailors on the Argo.