Welcome to the forums.
I don't see anything wrong with "Good Stay." The grammar is fine - "stay" is a noun and you want people to have a good one.
I'm not feeling very creative right now, but you could use what you have: "We want you to have a Good Stay." I like it better than something like "StayRight" or "Easy Stay."
Is the software for the vacationers, or for the hotel companies? That would help with the creative ideas.
Student or Learner