This street, like most others in Fort Lawton (1900−2011), which is now Discovery Park, was named by the United States Army after a state of the Union. I am not sure when the post’s streets were so named or who made the decision, but it can have been no later than 1944, when this map was made by the Army Corps of Engineers. (Here’s a much higher-resolution version from the Seattle Municipal Archives, created in 1973 but based on the older map.) One can see there that what are today Illinois Avenue and Bernie Whitebear Way were originally Vermont Way, Illinois Street, Lawton Road, and Florida Avenue. At some point before 1967 (see this map made by the Fort Lawton Office of the Post Engineer) the four were combined, and in 2011 the middle portion was renamed after Native American activist Bernie Whitebear.
Today, the eastern section of Illinois Avenue begins at Discovery Park Boulevard and goes ¼ of a mile north to Texas Way, where it turns into Bernie Whitebear Way. (Except for 45th Avenue W, no street in Discovery Park carries directional designations, nor did they when it was still a fort.) The western section, which is closed to traffic, begins at Texas Way and Discovery Park Boulevard and goes ¼ mile north to connect with footpaths that themselves connect to the North Beach Trail.
Born and raised in Seattle, Benjamin Donguk Lukoff had his interest in local history kindled at the age of six, when his father bought him settler granddaughter Sophie Frye Bass’s Pig-Tail Days in Old Seattle at the gift shop of the Museum of History and Industry. He studied English, Russian, and linguistics at the University of Washington, and went on to earn his master’s in English linguistics from University College London. His book of rephotography, Seattle Then and Now, was published in 2010. An updated version came out in 2015.