LORELAI: Oh no, it’s that arcade game where the mole keeps sticking his head out and you have to pound him as many times as you can with the mallet. You would be a master at that game. … They would erect a statue of you next to it with perfect hair and pearls and a big bronze mallet.
Lorelai is referring to Whac-a-Mole, a popular arcade game first introduced in the mid-1970s. A Whac-a-Mole machine is a cabinet with five holes, from which the moles pop up at random, while the player has to hit them with a large plastic mallet.
Lorelai says that Emily would be a master at Whac-a-Mole as she’s so good at continuing to “whack” at Lorelai, but in colloquial use when people refer to a situation being like Whac-a-Mole, they mean that it’s futile as the moles keep popping up annoyingly no matter how many times you hit them. Lorelai is unwittingly (?) providing a testament to her own resilience and cockiness in the face of Emily’s assaults.