When you say that you "would rather" (meaning "prefer") another person did/had done something, you have to use the simple past tense (stayed) if you want them to do that in the present or in the future. You have to use the past perfect (had stayed) if you wish they had done that in the past, but they didn't.

i think we use this expression for a contradiction for example : you wanna go out ? i would rather stay at home.