Are you sticking purely to ghosts, or ghosts-and-aliens-and-chupacabras-and....?
As I said, I don't actually have any plans to actually implement this in the short term, it's just an idea...
If I were to implement an actual game I would probably include a large range of different "events" in the game -- various types of spirits/ghosts and aliens seem to be the most easily explained and would work in a variety of settings with different types of feedback for players. Anything with a supposedly physical manifestation isn't really suitable, as we can't put actual objects in the world -- but we can simulate all sorts of visuals, mock radiation signatures, audio "at frequencies beyond normal human hearing", etc.
We could give them radiation detectors, a number of different camera modes, some sort of special microphone, EMF meters, and then have all sorts of different activity for different events and/or different stages of a given event.
We could also have more general detections warn the player to look for something more specific in the area -- consider an example where a player is just walking down the street, perhaps not even thinking about the game at the time -- we could alert them with a beep and/or vibration and on-screen-note telling them that "electro-magnetic fluctuations have been detected in your area -- keep an eye out for paranormal activity". If they're busy they can just dismiss the message and continue about their business, but if they're interested they could open up the app and try investigating or keeping an eye out using the other "tools" at their disposal.