I read this story (must have been in an anthology; I didn't read SF/F magazines) when I was a kid, which puts the publication date before 1975. I remember it as being much older than that, though, not Golden Age, maybe the 50s or 60s. In the story, the expedition isn't (I think) to hunt dinosaurs. I remember a savanna, so there was probably hunting of some sort involved. A member of the expedition does something rash or stupid that results in the death of the narrator's brother. I think the last line, or close to, is the brother's name fading on the wind as he's erased from existence. I've been all over the web looking for a reference to this story without finding it, which means it isn't well-known, although I think the author is.
Richard Marsten was born Salvatore Lombino. He was a prolific writer, and Richard Marsten was only one of his many pen names. As Evan Hunter, he wrote the best-seller The Blackboard Jungle, as well as the original screenplay for Alfred Hitchcock's The Birds. He also authored the original police procedural series, *The 87th Precinct. He is also know as Ed McBain.
A confirming synopsis can be found on the Winston Science Fiction blog. Many thanks to D.W. Rowlands and the people at the MIT Science Fiction Society for getting this information for us.
This short story was also performed as a radio drama in the mid 1950's, but it was not done on X-1, and I have been unable to find it anywhere. The plot centered around the idea that the two brothers ran time travel safaris for wealthy people to photograph dinosaurs. On this particular trip, the wealthy patron smuggles a gun into the time machine, and, when they arrive starts shooting at dinosaurs. In the ensuing havoc, over the course of several days, various members of the expedition are killed, and, as each one dies, his or her memory is slowly erased in the minds of the survivors, and history is altered to accommodate the fact that these people died millions of years before they were born. Eventually, the patron is killed, and history changes to make his servant, a black man, the eventual patron. (Very unusual for that era.) In the end, as you stated, the older brother is killed, and, in the final scene, the younger brother calls out his name in a futile attempt to retain his memory. it was the best time travel story I've ever run across. I hope you find it and let us all know where it can be located.
It was not on Dimension X or X-1. I have listened to all programs on both series, and have not found this story. I think I have also heard every program broadcast on 2000 Plus. About all I can add is that heard this story on a radio station in Boston, at night, during the school year, most likely during the school week, and, also most likely, between September of 1954 and June of 1956. (The story had to have been published at that time or before. The fact that it was a high quality radio production, does not necessarily mean it was a high quality short story, but, based on the radio production, I would guess that it originally appeared in one of the high quality SF pulps.) Either the name Orren, or Owen, as was mentioned above for the older brother that died, rings true in my memory, as does the idea of a savanna. I also think that the older brother had either a wife or a fiance, (she may or may not have been part of the safari crew), and that the younger brother, who idolized his older brother, was conflicted because of his own feelings for the woman, and that part of the history adjustment that takes place is that, in the end the woman is the younger brother's wife or fiance. He also winds up being the leader of the expedition. A lot of the elements that appear in this story, appear in later time travel stories like A Sound of Thunder. Our local NPR station runs OTR programs from 9pm to midnight six nights a week, and has done so for all of the 25 years I have lived here. In addition to X-1, Dimension X, and 2000 Plus, they run Escape, Suspense, and just about every other adventure and drama series ever broadcast during that era, and I have never heard them broadcast this story.