Similar, but not really the same.
I may well stay at home is like I think I just may decide to stay at home if the whim strikes me; in fact, there's a good chance that I will stay at home.
I may as well stay at home is like There's no point in doing anything else; it will be equally good for me to stay at home.
I had better stay at home is like It's to my advantage to stay at home because something unpleasant could happen if I don't.
CJ