Arwen wasn't Half-elven, and she didn't choose to be an Elf at all. Her father Elrond was, but he had made the choice -- granted by the Valar, through Illuvatar's discretion (hmm... well read the Sil first. I won't go into that [img]smilies/smile.gif[/img]) -- long ago to be an Elda; Arwen's father and mother were therefore both Elvish. Her fate would have been that of the Firstborn as well, had she not married Aragorn and adopted the destiny of Men.
Arwen certainly had human ancestry, but she was an Elf herself.
And no, it wouldn't have mattered if she hadn't had that ancestry. It is irrelevent. Idril was of fully Eldarin ancestry, whom Tuor wedded in the First Age (Tuor taking up immortality in Arda, and the Elvish and as Areruthiel mentions, Luthien`s (wife of Beren, Man, also in the First Age) parents had no trace of Mannish blood in them either.
The thing is, you see, the three Unions of the Two Kindreds had nothing to do with the ancestry of those involved, but with which kindred they themselves were. The different fates of Elves and Men are what counts principally in all. (You`ll probably get a good idea of this from the Silmarillion, though. Good luck!)
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