I found Americans usually say, "come on in", when they invite people to go into their houses.
I wonder how native speakers except Americans say this. Do you say 'come in' or 'come on in'?
It's not just Americans that say this, though I would agree that it may well have been American in origin. The phrasal verb 'come on' is (can be, in one of its meanings) an invitation - not necessarily a physical one: 'Come on... Someone must know the answer... Anyone?...'
So 'come on in' means 'come in' but with an extra sense of encouragement. An extra 'on' can be introduced into other phrasal verbs; a gameshow presenter was famous for the catch-phrase 'Come on down', inviting contestants down onto the stage; 'Come on along and listen to the lullaby of Broadway' (a song - Come on along and listen to ); 'Why don't you come on round some time?'...