ST. LOUIS — Die-hard music lovers, casual fans and globe-trotting tourists readily embrace Chicago as the home of the blues, spurred in large part by the Great Migration northward of Southern blacks in the early and mid-20th century.

Robert Johnson, the genre’s godfather, famously sang of “Sweet Home Chicago,” and the Chicago Blues Festival draws more than 100,000 people each summer.

But come next year, the National Blues Museum won’t find a home in Chicago, but in a rival Midwest city 300 miles to the south....