By "stay," Hades may have meant "die"
The full context:
Hercules: You like making deals. Take me in Meg's place.
Hades: Oh, well. The son of my hated rival trapped forever in a river
of death.
Hercules: Going once!
Hades: Hmm. Is there a downside to this?
Hercules: Going twice!
Hades: Okay, okay, okay, okay. You get her out - she goes, you stay.
It seems as the bargain Hades was talking about was for Hercules to be "trapped forever in a river of death," i.e. to die. Once Hercules became a god, of course, he no longer qualified for death, and so the conditions of the deal were no longer valid. Hades could have pressed the point, but with Hercules now a god, he probably could not keep him there. Besides, he really wanted a dead mortal Hercules, not an immortal Hercules following him around and making snide comments.
In addition, Hades broke his part of the deal
Hades never intended to let Meg go, as evidenced by the fact that it "slipped his mind" that Hercules would die before he could help Meg out of the water. Hercules may well have felt that there was no reason to keep his end of the bargain, since Hades had already reneged on it.
Hades: Okay, okay, okay, okay. You get her out - she goes, you stay.
[Hercules dives]
Hades: Oh, you know what slipped my mind? You'll be dead before you
can get to her. That's not a problem, is it?
[Hercules swims, turning older and older. Atropos goes to cut the
thread of life, but it suddenly shines and the scissors don't cut
it.]
This could also explain why Hades didn't try to claim Hercules later: there was no deal, since he had already gambled on getting both Hercules and Meg. Hades gambled, "all or none," and lost.