It is curious to note that Half-elven status seems to have descended differently through the lines of Elrond and Elros. Elrond's children shared the same choice as their father (and thus Arwen was able to choose mortality, though her father had chosen to be counted an Elf). Elros' children did not have the same privilege: his descendants all shared his mortality, though many craved the immortal life of the Firstborn.
Tolkien himself mentions this distinction in passing, but doesn't offer a definitive explanation: 'Elros chose to be a King and 'longaevus' but mortal, so all his descendants are mortal ... Elrond chose to be among the Elves. His children ... have to make their choices.' (The Letters of J.R.R. Tolkien, No 153, dated 1954).
The reasoning behind this is uncertain, but the explanation probably lies in the fact that the children of Elrond in Middle-earth had to make a choice eventually: either to depart across the Sea as an Elf, or to remain and die as a Man. There was no comparable choice facing the children of Elros and so, it seems, no need for them to inherit the power to choose.