This street was named after the Brooklyn Addition to Seattle, platted by James Alexander Moore (1861–1929) in 1890. The addition itself was named for Brooklyn, New York (see the history of the University District Paul Dorpat wrote for HistoryLink and this article on his own blog):
[Moore] chose the name because his addition “looked across the water” to Seattle proper like the New York borough of the same name that looks across the East River to Manhattan.
I haven’t been able to find an online source for this assertion, but Dorpat may have been referring to a passage in Roy G. Nielsen’s UniverCity: The Story of the University District in Seattle (1986), which quotes an article in the August 31, 1928, issue of the University Herald in which George F. Cotterill (mayor from 1912–1914) says:
[In 1885], there was no thought of a university and section sixteen was still untouched. J.A. Moore, one of the greatest Seattle real estate promoters of the time, started Brooklyn between 10th Ave. N.E. and the campus. This addition was intended by Mr. Moore to be to Seattle what Brooklyn is to New York.
Today’s Brooklyn Avenue is a block west of Moore’s. Again quoting Paul Dorpat:
None of James Moore’s street names survive. His Tremont Avenue became 15th Avenue. One block west he named University Way — the District’s future “Main Street” — Columbus Avenue. He called the future Brooklyn Avenue, “Broadway,” and this was Moore’s intended “Main Street.” He called 12th Avenue “Brooklyn.”
Brooklyn Avenue NE begins at NE Boat Street just north of Fritz Hedges Waterway Park and goes 1¾ miles north through the University District and Roosevelt neighborhood to NE 66th Street at Roosevelt High School. It resumes at NE 70th Street and goes just short of 300 feet to Froula Playground and the Roosevelt Reservoir. There is another segment between NE 75th Street and NE 77th Street and a final one between NE 80th Street and NE 82nd Street by Maple Leaf Reservoir Park.