The kanji (Chinese character) for Asuka is written with two characters, one meaning "to fly" and the other meaning "bird.". Some say it's derived from "of flying bird." (tobu tori no).
In Japan, there was a period of time which people now call the "Asuka Period." The Asuka Period started towards the end of the 6th century and lasted about 100 years. During this time, art and architecture thrived and Japan's capital was in the Asuka region. In the Asuka period, currency was issued and architecture and art were making major advances. It was a very successful time for Japan.
The word's literal meaning "to fly" and "bird" are both signs of being up high, like the period of time mentioned.