This street runs nearly 6 miles from the north end of the University Bridge in the south (at Eastlake Avenue NE and NE Campus Parkway) to Aurora Avenue N in the north, just shy of Seattle city limits at N 145th Street. It runs north–south for most of its length, but starting at NE 125th Street, its last 1½ miles cut a northwest–southeast diagonal across the street grid, making it Roosevelt Way N once it crosses 1st Avenue NE between N 133rd and N 135th Streets.
Originally 10th Avenue NE south of NE 125th Street, Roosevelt received its current name in 1933. According to local historian Feliks Banel, this was first proposed in 1927 by businesses in the Roosevelt district, itself having taken that name earlier in the decade in honor of President Theodore Roosevelt, who died in 1919. Nothing came of it for six years, but in 1933 they tried again and asked that 10th Avenue be renamed after both Theodore and Franklin Delano Roosevelt, who had been elected president the previous November. As Banel notes in his piece, The Seattle Times had this to say:
This change, we assume, must be pleasing to local Democrats of all sorts and shades. At the same time, due to the incidence of somewhat tenuous family ties, it cannot be at all displeasing to Republicans. The name of Roosevelt has high standing in both political parties, as indeed it has throughout the world. Even now it is quite certain that those who may traverse our Roosevelt Way in years to come will neither know nor particularly care whether it was named for Teddy or for Frank.
As for its diagonal stretch, it appears on old King County maps as M. Roy Sayles Road (County Road 2240), Golf Way, and State Highway 1J (predecessor of today’s SR 513). It ceased to be a state highway in 1991. As for when it, too, became Roosevelt Way, it’s difficult to tell as King County doesn’t have as good a system for looking up ordinances online as Seattle’s. It appears as Roosevelt Way on a 1966 map in local historian Rob Ketcherside’s maps album on Flickr, but as Golf Way in another one from 1947. As the area in question wasn’t annexed into Seattle until 1953, the name must have been changed by the county sometime between 1947 and 1953 in anticipation. (The 1933 Seattle ordinance is the only one on file relating to Roosevelt Way’s name, so this must have been a county change.)
As for M. Roy Sayles, The International Confectioner’s January 1915 issue reports that he, along with Annie B. Sayles, C.M. Sayles, and W.H. Rogers, founded the Rogers Candy Co. in Seattle in 1915; and Golf Way almost certainly comes from the road’s proximity to the public course at Jackson Park, which opened in 1928.