I was thinking of some mini games that would fit the noir theme and would also be fun. I feel like a 70s/80s boxing gym fits. It would be a new building type, a small boxing gym where you engage in fights where only fists are allowed. It could also perhaps be an underground fight club or a combination of both. I feel that fits in with the setting as well.
You unlock it by having a certain reputation, paying a joining fee/ membership, or if it's an illegal fight club, you need a password to get in that would take some detective work to discover.
There would be progression where you start off with easy opponents but work your way up to fighting the champion. At a certain point the opponents you fight there would be tougher than the most normal NPCs you encounter in the city.
It would be a fun way to pass the time in game but also learn and get good at the combat mechanics. This idea should come with an overhauled / more interesting combat system.
Some extra ideas inspired by the comments:
Fights would be happening even when you're not there. There'd be a fight record somewhere that shows fights for the past 2 days, kind of like sales records already in the game. Would have the initials of who fought and who won. This could of course benefit investigations when looking for someone and there could even be cases specific to the fight club, like a murder occurs and it seems that the culprit beat the victim to death with their fists, leading you to investigate the fight club. Or a fighter asks you to investigate their upcoming opponent to find weaknesses by bringing them a file of their opponent. That could bring in things like medical records already in the game.
You could bet on other citizens’ fights as an extra way of making money. The game could choose random NPCs who participate and it's part of their schedule on some days. Could be flyers with the date/time of a fight, or certain citizens’ calendars will have a fight marked on it.