The "big brass ring" of the title is a reference to the high brass rings found in old fashioned gyms and swimming pools, which one was supposed to jump up to and grab. It is a metaphor for that which is difficult to attain, or at least hard to hold fast and retain.
This movie contains numerous Shakespearean references, including direct quotes from the Bard's plays.
George Hickenlooper and F.X. Feeney heavily re-wrote the screenplay and made substantial changes from the Orson Welles script (published in 1991 by the University of Santa Barbara Press). Welles' script concerned Senator Pellarin, a Democratic Presidential candidate in 1984 (closely modelled on Senator Gary Hart), and his troubled relationship with his disgraced homosexual mentor Kim Minnaker, a one-time Roosevelt New Deal Democrat, who was now living in exile as advisor to the corrupt government of an unnamed African dictatorship. Hickenlooper retained the basic concept, but instead re-cast Pellarin as a candidate for Governor of Missouri, and Minnaker as living in Cuba, while much of the dialogue was re-written and re-interpreted. None of Welles' satire of Reagan-era politics was retained in the final movie, while several key scenes, like a charged confrontation between Pellarin and Minnaker on a Ferris wheel, were also omitted.
According to co-writer F.X. Feeney: "On the surface, it is a very free adaptation. Underneath, it is highly faithful. Orson Welles' original script was set in Spain and the Congo. We set ours along the Mississipi and in Cuba. Nevertheless, the characters have kept their original names and essential personalities through the many adaptations George Hickenlooper and I devised (whether separately or together) between 1991 and '98."