Man, who'da thunk that such an ugly piece of kitchenware would cause such a fuss? If we were in the bribe-giving game, we would have chosen something way prettier.
In fact, when More goes to Wolsey's and receives the cup, he doesn't know it's a bribe. He only realizes that it's a bribe later on, as he's traveling down the Thames back to his home. (The cup-giver, Averil Machin, wanted him to grant her a favorable decision at a property dispute in court.)
He almost dumps it in the river, but the boatman stops him, saying that it's worth money. Instead, he ends up presenting the cup to Rich, using it as an example of the kind of temptation that a person seeking a place in court needs to deal with. He tells Rich it would be better for him to go with an academic life, and presents Rich with the cup as a reminder of the corruption of public office.
Rich uses the cup against More, telling Cromwell about the supposed bribe. But Cromwell's attempt to smear More and accuse him of corruption falls flat when the Duke of Norfolk points out that the dates don't match up—More got rid of the cup before he decided the case. Cromwell has to resort to other, more devious tactics in order to pressure More, eventually arresting him for high treason for not signing the oath on the king's marriage.
So this ugly, tarnish-prone cup symbolizes the corrupting effects of worldly power. It's fitting that it should pass out of More's hands and into Rich's, since Rich is the one who will prove extremely susceptible to the seductions of worldly power… while More can easily dispense with them.