J. Lyman was a Starfleet officer assigned to the starship USS Voyager, a lieutenant who was killed in the line of duty sometime before 2377. Lyman's name was read out on a casualty report of all who died during that vessel's journey through the Delta Quadrant. (VOY: "Imperfection")
This name was devised for the list; it was not mentioned as a casualty in any other episode. It is possible this name refers to one of the officers killed en route when Voyager was dragged to the Delta Quadrant. As with many names on the "Imperfection" list, Josh Lyman is an in-joke referring to the character on The West Wing. Josh states that he is a fan of Star Trek in the episode "Arctic Radar" of The West Wing.