Story Elements
Chicago had a lot of gangsters in it's time. Maybe one of the things, similar to New Vegas' casinos is to stop the crime of the gangsters, join one of the families, or make your own, whether it be lawful or chaotic.
Brotherhood of Steel is very heavily-occupied in Chicago I believe. Maybe they can be at a drastic struggle of trying to stay out of the gangster's path in order to not be obliterated by all of the families.
ED-E said something about how there are Enclave in Chicago also. Enclave could be another big part, probably taking Chicago for themselves, but having a hard time with the BoS almost dominating the territory.
Parts of Chicago I would love to see:
Buckingham Fountain
Why not take a drink from an irradiated Buckingham Fountain? Michigan Lake Mirelurks could take refugee here because of the rainforest-like aroma that the fountain gives off.
Chinatown
We've never seen anything like this before. Here, the hero could encounter Chinese terrorists who planned to bomb Chicago during the war. On the topic of gangs, Chinatown could see a Yakuza-inspired gang.
Drake Hotel
A well-known Chicagoland building and hotel. Could be the headquarters for a gangster family.
Michigan Avenue Bridge
One of the main objectives of Fallout: Chicago could be going to the southside of Chicago. The Michigan Avenue Bridge, according to wikipedia, was once the main link between the north and south side of Chicago across the Chicago River. Maybe the protagonist needs to find a way to put the drawbridge down again in order to get across.
Millennium Park
This could be the main battlefield between the two main factions. Fallout 3 had the mall and New Vegas had the dam. Chicago can have the park.
Navy Pier
I'd love to see an apocalyptic Navy Pier. Fallout 3's DLC, Point Lookout, basically had it right, in the carnival broadwalk section. Could be one of the most dangerous parts of the Chicago Wasteland, having everything from raiders, gangsters and mirelurks. One of the buildings could a warehouse where a classic sneak into the warehouse to stop the gangster drug deal or save the president daughter mission is held.
Sears / Willis Tower
Once the tallest building in the world. Maybe they can have it where a nuke obliterated the top of the tower, and caused it to crush a good portion of the surroundings. Although this may seem very controversial with 9/11 and all, Fallout 3 did have the White House turned to nothing more than dirt.
Tribune Tower
One of the most important buildings in Chicago. Could be the Chicago BoS' citadel.
Wrigley Field
A very well-known baseball field for the Chicago Cubs. Could be overrun with Raiders who have equipped themselves with baseball equipment. Already have an idea for a schematic: a baseball automatic pitcher. Maybe there could even be a reference to the goat curse.
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Gameplay Elements:
Windy City
Chicago is known as the Windy City. Now, I don't think the wasteland should be called the Windy Wasteland since that sounds stupid. I'm just going to refer to it as the Chicago Wasteland. Anyways, a common element, especially for the less city parts, could be a deadly duststorm, similar to that big crater in New Vegas.