According to two websites, one an Evangelion fan site, the other a random one, Asuka means "tomorrow fragrance," from asu, tomorrow, and ka, fragrance/perfume. Someone else said it was "tomorrow beautiful," assumably interpreting ka as "beautiful." And others have said that the kanji of all the NGE names are unavailable, so the exact meaning could be unfindable.
[Another site, non-Evangelion, lists Asuka as a place name written with character meaning "to fly" and "bird." "One hypothesis is that it derives from the stock epithet "of flying birds" (tobu tori no) commonly prefixed to the place name 'Asuka' in poetry." Probably unrelated, but interesting. ^_^]
And I'm convinced that if names have secret, hidden meanings, Asuka's is "Anta baka?!" ;D
Getting to the point ---> "Tomorrow fragrance" is probably your best bet.