One important point about the three dominant characters is their names: Sophy, Mr. Twycott and Randolph Twycott. Sophy is a name from the Greek meaning 'wise' or 'wisdom'. This is very important to know about Sophy because Hardy is indirectly stating that Sophy's choices cannot be faulted at any point because she made wise choices. He is effectively removing any thoughts of fault or blame for her situation from off Sophy's shoulders.
"Twy" is an Old English word meaning 'two'. "Cott" is an Old English attribute name given to people who were hard-hearted, unfeeling person and is related to a person who makes hard chain-mail, a hard material for warfare. The symbolism here is that Mr. Twycott is a hard, unfeeling person who lives as though going into warfare. This is borne out in the story because (1) he does not teach their son to respect and speak kindly to his mother though she may be from a rustic background and (2) that upon marrying, he removed Sophy from her village by taking a post in the outskirts of London where the society they'd associate with would not be shocked by her rustic ways.
The name Randolph means 'shield' or protector. This is already painfully ironic without delving any further into the symbolism. Randolph absolutely did not shield his mother; he bullied and manipulated her. Yet there is more to his name than this. He is the other half of the "twy + cott" duo. He is the second of two people who are hard-hearted and unfeeling.
An important theme that emerges from a discussion of their names is that not even Wisdom can protect itself against hardened hearts and unfeeling warring manipulators. This is a thought provoking universal theme since we all aspire to wise action (or should do) and, it seems, we may all be prey to hardened unfeeling hearts, just as Sophy was. Unfortunately, Hardy doesn't present an optimistic antidote to this problem: Sophy dies and is marched past bereaved Sam while Randolf sits in authority over her (even in her funeral train) like the black cloud of a jailer or executioner.
[Sam], whose eyes were wet, held his hat in his hand as the vehicles moved by; while from the mourning coach a young smooth-shaven priest [Randolph] in a high waistcoat looked black as a cloud at the shop keeper standing there.