The motif of digging is referring to uncovering that which was hidden or lost. It begins in the first book, "Recalled to Life," in which Dr. Manette is released from prison. Long held in captivity, Manette has lost most of his reason. The process of "digging" to find his sanity is left to Lucie over the years. Yet still, he has occasional bouts of relapse, reliving that past which he believed to be buried.
In the same way, Darnay is forced to reveal a past which he too believed to be buried. His connection withe the Evremond family has been rejected by him, but in his arrest and trial, it must be uncovered.
Jeremy Cruncher's "sideline" of a graverobber is symbolic of this process. He sells corpses to medical students and doctors.
In all, each character has something "buried," which must be dug up and faced. All secrets must be revealed, for good or ill.