This street — named Pine Street in Terry’s Second Addition to the Town of Seattle, filed in 1872 — was renamed Terrace in 1876, presumably to avoid duplicating the Pine Street to the north. My assumption is it was given this name because of how steep First Hill is at this point — Paul Dorpat, writing for HistoryLink, says “Except at Terrace Street on the south and near Union Street on the north, the hill was readily negotiable first by hikers and later by street graders and trolley tracks.” One can see in the image below the long, steep stairway in the Terrace Street right-of-way “from 5th Avenue east to beyond 7th Avenue,” as Dorpat writes on his own blog.
Today, Terrace Street begins at Yesler Way above 4th Avenue and goes a block and a half up the hill before being blocked by Interstate 5. It resumes just east of Harborview Medical Center and goes ⅓ of a mile east to 12th Avenue. There is another short stretch of E Terrace Street from 20th Avenue to 23rd Avenue by Garfield High School, and then a final ½ mile from 30th Avenue to Lake Washington Boulevard in Leschi, the portion between 36th Avenue and Randolph Avenue being a stairway.