A cosmonaut? Hmmm...while that is a logical connection to the team name, wouldn't a cosmonaut more be identified with the hated Soviet space program (a fierce rival with the United States in the Space Race), and not involve Chicago?
Hey...I saw Chicago native, soccer fan, and film and screen Star Jim Belushi at a Fire game at Toyota Park last season, is he available to serve as the mascot?
Perhaps invite star Chicagoans to serve as honorary captains before each game? Stars appearing at the games would certainly get media attention. A potential Chicago Red Stars Honorary Captains List: Joan Cusack, Jenny McCarthy, Gillian Anderson, Jennifer Kate Hudson, Frances McDormand, Hillary Rodham Clinton, or maybe, John Cusack, Harrison Ford, Bill Murray, Jim Belushi, John Malkovich, William L. Peterson, Gary Sinise, John C. Reilly, Robin Williams, Dan Castellaneta (Homer Simpson), Vince Vaughan. Please add additional names.
Something that represents feminine strength, but also represents the backbone of Chicago, the working class. Maybe something along the lines of Rosie the Riveter (with blue overalls with the 6 pointed red star on it). Don't know how you can translate that in to a mascot, but just an idea.