The designated dave/driver is the person assigned to transport a group of drunk friends to and from a party. It's considered a shitty job because:
a) You're sober
b) You have to deal with drunk people while you're sober
'Designated Dick' exists because any Tom, Dick or Harry can be chosen to fulfill the role and being chosen or being 'forced to volunteer' feels like an unwelcomed blap to the forehead.
(Friends sitting in a room together)
Person A: So who's all going to the rave on Friday?
Person B: Me, like fuck!
Person C: Me
Person D: Me, I'm going to get so wasted.
Person E: Me
(Everyone turns to Person E)
Person A: Isn't it your turn to be designated dick?