If I'm trying to introduce someone to anime my goal is to wow them first before I get them into all the strangeness of anime, because I know from experience that the if you jump into some anime's it can cause you to not want to watch it. so I try to go by their taste in movies. If they have a darker taste I would go with either death note or darker than black.
If they like romantic comedies I would go with the host club or toradora. Both of which can be a little on the stranger side.
or if they want something with a little romance and darkness I would go with vampire knight. I don't have one specific choice. I try to pick something I know they will like.