RICHARD: The girl [Rory] obviously needs some peace.
EMILY: How do you know that?
RICHARD: I can tell.
EMILY: Oh, you’re a mind reader now, how nice. We’ll get you a turban and a little booth by the train station.
Emily may be referring to coin-operated fortune telling machines, which display an animatronic figure in a glass booth which dispenses fortunes, either on a little card, or in a recorded voice. Her description of a turbaned figure sounds rather like the Zoltar fortune teller which appears in the 1988 film Big, starring Tom Hanks. Although created for the film (and based on the real-life Zoltan fortune telling machines), since then there have been Zoltar machines made to resemble the one in the movie.
Fortune teller machines are usually in amusement arcades or at fairgrounds, but in times past there were sometimes machines designed to tell your fortune at railway stations (often combined with weighing you at the same time). No doubt the idea was that people hanging around waiting for trains had time to kill, and spare change. These kind of fortune teller machines are still widely available at train stations throughout Asia.