We don't know much about Herbert White. How old is he? Why is he still living with his parents? How did he deal with the death of his siblings (whom we know even less about than him)? What does he do for work? How exactly did he die?
Sorry, we can't answer any of these questions for you, but they aren't that important to the story. What is important is that Herbert seems to die senselessly before his time, leaving behind grieving parents. What's also important is that Herbert is obviously very loved by his parents. Even before he dies we can see that he gets lots of love and admiration from them. What's more, he seems to care about them just as much.
We get the idea that he works hard and is responsible, cheerful, smart, and imaginative. He values his parents and enjoys spending quality time with them. He is probably the only one working in the family, supporting them all. Basically, he's a model son – every parent's dream. His good qualities make us mourn his death and feel for his parents when they lose him.
Herbert seems the least likely character to believe in the power of the monkeys paw, but he's the one who sees (or thinks he sees) the monkey's face in the fire. Like his parents, Herbert both wants and doesn't want to believe in the paw.
Herbert, in his normal human form, disappears from the story fairly quickly, but the idea of him haunts the story after his death. First he's mangled up in a machine at work so badly that his father can barely look at his body. Then he is wished back to life and forced to walk (all mangled and rotting) the two miles home from the cemetery. When he gets there, his parents won't even open the door at first, then his dad wishes him back to death. Wow. Hopefully Herbert can get some rest now – he deserves it.
On a very seriousness note, Herbert's death might be a nod toward the reading public's horror at the fact that many people, especially men and boys, were regularly killed or injured in factory accidents in England in the early 1900s. By focusing on the horrors Herbert's body undergoes, this story honors and memorializes those real victims.