I don't know whether the stone given to Frodo by Arwen actually had a specific name, although maybe we're directed to think that it *is* the Evenstar, because when she gifts it to Frodo, Arwen says:
Quote:
... But wear this now in memory of Elfstone and Evenstar with whom your life has been woven!
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And the stone itself is described as:
Quote:
... a white gem like a star...
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And, of course, Aragorn has his Elessar (Elfstone) given to him by Galadriel, and he then takes this as his name, so maybe we
are supposed to think that the gem is the Evenstar, linked with the name of Arwen herself. But to my knowledge, it's never mentioned as such. (But hey, what do I know? [img]smilies/biggrin.gif[/img] )