The 45-year-old man's name is Henry. In the first half of the story, his name is unknown, and he is referred to as "the man." We only learn of his real name when another miner named Tom drops by to hear of news of Henry's wife. It is then that Tom calls Henry by his name.
As for why Twain does not immediately disclose the man's name, we might assume that this mirrors the mystery of the character. The narrator does not really understand Henry until the very end of the story. For the reader as for the narrator, Henry's secret - that he has gone mad after his wife's death - is not revealed until the very end. Henry is thus a mysterious character who we only get to know bit by bit.