Specifically because he left them a lot of money and he told them exactly what to name it in his bequest.
Per their website:
"The Smithsonian Institution was established with funds from James Smithson (1765-1829), a British scientist who left his estate to the United States to found "at Washington, under the name of the Smithsonian Institution, an establishment for the increase and diffusion of knowledge." On August 10, 1846, the U.S. Senate passed the act organizing the Smithsonian Institution, which was signed into law by President James K. Polk.
Congress authorized acceptance of the Smithson bequest on July 1, 1836, but it took another ten years of debate before the Smithsonian was founded. Once established, the Smithsonian became part of the process of developing an American national identity-an identity rooted in exploration, innovation, and a unique American style. That process continues today as the Smithsonian looks toward the future. "
https://www.si.edu/about/history
