This street, which runs about ¼ mile from Lake Washington Boulevard E to the beginning of the Foster Point Trail, all within the Washington Park Arboretum, was without a name until 1968, when it was named for the island in Union Bay to which it led. (It remained unsigned until a few decades later, however. There was no sign at the intersection until at least the 1990s, as I know since my parents’ house was at the south end of the Arboretum and I drove or biked by there weekly, if not more often, while I was growing up.)
Foster Island is known by the Duwamish tribe, who once used it as a burial ground, as Stitici, or ‘little island’. It was named by the settlers for Joel Wellington Foster, who came to Washington in the 1870s from St. Joseph, Missouri. He is said to have donated the island to the city in one HistoryLink article, but another says the city bought it in 1917.