Bub is Malory Archer's mother, the maternal grandmother of Sterling Archer and the great-grandmother of Abbiejean Kane-Archer. She is never seen, but she is referred to in Job Offer during a conversation between Malory and Len Trexler.
Woodhouse held Bub's ring for the Archer family. When Sterling Archer professed that he intended to marry Katya Kazanova, Woodhouse gave Archer Bub's ring for the wedding.
In one episode, Malory replied that she had good news to which Sterling asked if Bub finally died but Malory gave him a cold look and said that wasn't the news. From Archer and Malory's tone it is implied that Bub is still alive and her relationship with her daughter might be on the tense side based on the comment her grandson had made.