[the phantom]

Burrahobbit is right, elves are not known to reuse names, but I also like Galorme's theory. Bear with me now.
Remember, when Turgon and his people began the pilgrimage from Neverast to Gondolin, it was said that he had a greater following of Sindar than his own people. If Thranduil was indeed around at that time in Doriath or around there anyway, it is entirely possible that he was one of the Sindar that went with Turgon to Gondolin at the beginning, and Legolas was born there. Or, perhaps Legolas was already born and he himself went to Gondolin.
This would explain his quick recognition of the balrog, and also his less than amiable nature towards dwarves, since they attacked his people in Menegroth, where perhaps his father lived.
I'm sure Legolas had heard from Thranduil hundreds of times throughout the years "Did I ever tell you about when those filthy dwarves sacked our home?"
"Yes, dad, you told me that just last millenia."
Anyway, I never imagined Legolas being that old, but I don't think we can rule out the possibility, not yet anyway.
Plus, in the books, I don't think it ever said how old Legolas looked, or appeared to be. Maybe Tolkien imagined him being older, and would've been very disappointed with the choice of Orlando Bloom, because he was too young.
In Fangorn, Legolas said he almost felt young. Well, if he was indeed born in the third age, you'd think he would've definitely felt young! If he was born in the third age, he'd feel young every time he was around other elves! It wouldn't be such a novel experience for him to feel young!
Anyway, that's my two cents.