The Book of Ether is largely the record of the Plates of Ether. However, Moroni is upfront about the fact that he omitted details recorded therein, covering the time from the creation to the tower of Babel. Hence it is the portion which gives a history of the Jaredites. Nevertheless, it is clear that in Moroni's editorial capacity, he glossed over some details even in that history. For example, he mentions that the Brother of Jared moved Mount Zerin by his faith, but this story is not anywhere related in the Book of Ether. It is worth considering, perhaps, that the Record of Coriantumr may have been included in the Lost 116 Pages, and that, inasmuch as Moroni makes reference to the story as if it was something he expected his readers to be familiar with, perhaps the story was told there, and then not repeated in the Book of Ether. This might explain why the Book of Ether seems to leave so many people unnamed as a peculiar feature of the book, and yet Joseph Smith, Jr. still seems to know the name of the Brother of Jared. (Though the story we have of Joseph Smith sharing the name of the Brother of Jared says that it had just then been revealed to him.)
The Book of Ether is replete with characters, some of them quite important, and even a place, which go strangely unnamed. The Brother of Jared is only the first and most glaring example.
Since Joseph Smith knows the Brother of Jared's actual name, it suggests that his actual name was on the plates. However, since this only seems to affect the Book of Ether, it suggests that this is either a feature of that record and its record keeper, Ether, or a feature of Moroni's unique style in summarizing the history. (This is the only book Moroni summarized.)
In order of appearance: