Just as with E North Street, one question that should spring to mind with a street name like this is Eastern side of what? — especially considering that it’s not too far from being equidistant from Lake Washington and Salmon Bay, measured along the east–west axis of 50th Street.
Local historian Rob Ketcherside explains in his blog post on “Renaming streets of Seattle’s Fremont to the U District”:
It was originally named East Street in the 1883 Lake Union Addition plat.… it was the street most east in the addition. That plat… [was] annexed into Seattle in 1891. Sometime apparently after 1893 and before 1895 East Street was renamed Eastern Avenue.… The streets running perpendicular to Eastern Avenue are labeled North until one block east. Then they are labeled Northeast. The first avenue east of Eastern is First Avenue Northeast. So the streets east of First Avenue Northeast are Northeast as well. Eastern Avenue and the avenues west are North.… Say that ten times fast.
So — Eastern Avenue is the easternmost avenue in the section of town where east–west streets carry the N prefix and north–south avenues carry the suffix N, though the actual dividing line is a block east at 1st Avenue NE.
Eastern Avenue N begins on Lake Union at Waterway 17, just south of N Northlake Way, where it essentially functions as a parking lot for the restaurant Westward (much like 5th Avenue NE, a few blocks to the east, does for Ivar’s Salmon House). It begins in earnest on the other side of the Burke–Gilman Trail at N Pacific Street and goes ⅞ of a mile north to N 50th Street.