Didn't we do that already, but let them play our home game at the United Center?
We did all four:
2009-10: Gonzaga 85, Illinois 83 in OT (Chicago, IL)
2010-11: #20 Illinois 73, Gonzaga 61 (Seattle, WA)
2011-12: Illinois 82, #19 Gonzaga 75 (Champaign, IL)
2012-13: #13 Illinois 85, #10 Gonzaga 74 (Spokane, WA)
I would be SO in favor of more series' like this. Stagger them so you always have a United Center game, but keep each series to four locations over four years. There would be some great options out there:
Arizona: Champaign-Tucson-Chicago-Phoenix
Arkansas: Champaign-Fayetteville-Chicago-Little Rock
Florida: Champaign-Gainesville-Chicago-Orlando
LSU: Champaign-Baton Rouge-Chicago-New Orleans
Oklahoma: Champaign-Norman-Chicago-Oklahoma City
Oregon: Champaign-Eugene-Chicago-Portland
Syracuse: Champaign-Syracuse-Chicago-New York City
Tennessee: Champaign-Knoxville-Chicago-Nashville
Texas: Champaign-Austin-Chicago-Dallas
Texas A&M: Champaign-College Station-Chicago-Houston
Virginia: Champaign-Charlottesville-Chicago-Washington, DC
This leaves off obvious "dream series'" like Kentucky, Kansas, Duke, etc. A lot of people don't like neutral site games for some reason, but I really do. For one, it really does help build your brand in major recruiting hot beds and important markets. I don't believe for a second that our St. Louis recruiting/profile hasn't benefited from the Braggin' Rights Games or that our old games at the United Center that would constantly draw 16k or more were a nice recruiting pitch to Chicago recruits. Additionally, it gives fans a fun trip to some of the cooler/more touristy destinations on that list like, say, NYC or New Orleans.