This is an interesting question, and since the story is so short, we will have to read a lot into the little that is given to us. The best clues to answer this question are given in the first couple paragraphs of the story.
Looking closely, the story states that there had been a train crash on the tracks, and that a telegram had been sent to the newspaper office, giving the names of the people that had been killed. Brently's friend Richards had been in the newspaper office, and received the telegram. Here is where the clues come in. Clue number one is when Chopin writes that on the telegram, "Brently Mallard's name [was] leading the list of 'killed'". So, on a list of people who died in the accident, his name was put first. This could imply that he was important in the community, because his name made the top of the list that was going to the newspaper. Newspapers want headlines, and a prominent citizen dying would make more headlines that someone that the community didn't know. So, it was "newsworthy" that he would have died. That seems to indicate that Brently was indeed someone well-known.
The second clue comes from Richard's reaction to the news. He pauses only to send a "second telegram" to make sure it was true, then hastens over to the house to tell Louise. Even though there were many names on the list, and many families that needed to be notified, Richards runs over to Louise's to help to tell her first. This could be that Louise is simply a friend of his, and he is paying her a courtesy. It could be because they are affluent and special care is given to prominent members of the society. It could also be that he wanted to give her the news himself before it came out in the paper and she discovered it that way. She had "heart trouble" and reading of your loved one's death is not only insensitive, but it could've impacted her death.
Given the slight clues in the text, we can infer that Brently Mallard was indeed a noteworthy and well-known member of their town, one prominent enough to make headlines and to elicit immediate action from papers and friends. I hope that those thoughts helped; good luck!