RICHARD: I would hug you, but I have various forms of viscous fluid on my clothing.
RORY: I’ll take a rain check.
To “take a rain check” is a polite way to turn down an invitation, with the implication that you are only postponing it to a more convenient time. It originates from American baseball, when games would provide tickets to be used in future when games were postponed by rain, and dates to the late 19th century. Rory means she will hug Richard when he is wearing clean, dry clothing again.