Idril was 100% Elf and therefore couldn't give up anything.
The only reason Arwen (and Elladan and Elrohir) had a choice, is because they had mortal blood. Arwen chose to remain behind and not pass over the sea with Elrond and thus she became mortal.
Simple as that.
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-"Death borders upon our birth, and our cradle stands in the grave. Our birth is nothing but our death begun."
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