I'd make it a free-roaming game, but it wouldn't be constantly night. Nighttime would typically be the only time that plot missions pop up on the menu, and daytime makes stealth next to impossible for Bats. For the random crimes, split Gotham City up into territories.Street level crimes would be fairly typical, but indicative of the area. In the Barrows, you get more attempted rapes and muggings, while in the business sector, you have less crime, but more things like bank robberies and stolen cars. Fighting crime in a certain district would cause the fear level to go up in that area, and slowly reduce the instance of crimes. Occasionally, after stopping a street crime, you pump one of the thugs for information, or after completing a story mission, you uncover evidence that points you back to street crime in a different district. This would open up mini-missions such as crashing meetings of mob bosses, intervening in a gun deal, or shutting down a drug lab. There's two ideas that I think would be a cool fit for that as well. Infiltrating the criminal underworld as Matches Malone could be interesting. So is the possibility of calling the Gotham PD if a meeting is too big for Batman to control, but you have to keep the bad guys contained until the cops get there. That might be too ambitious though.
The plot missions would be cued at night. To start with, you'd have a number of leads to follow, which in turn lead to missions. The mechanics would shift between a long distance, pulled-back system for raising the fear in enemies and slowly picking them off from a distance, and an up-close brawler system. The missions would involve Batman stopping a large cast of his rogue's gallery in their own schemes, and eventually finding out that all the plots tie together (eh. sue me. I'm not writing a video game script here). Each mission you'd be given a grade on performance, and points to be spent on unlockables. I'd set up a sort of "Danger Room" which lets you go back into missions to get better grades and pick up clues you might have missed.
During the daytime, you'd be occasionally forced to go out and do a mission as Bruce Wayne, but the bulk of the daytime you'd spent back at the Batcave. Here, you analyze clues that you gather on missions (Plug in some mini-games to take care of that). Here (and maybe from the Batmobile), you can access the bat-computer database which would have all sorts of nifty goodies for fans. Things like Rogue's Gallery profiles, a history of Batman, design work, that kind of thing. From the Batcave, you can also allocate WayneTech R&D resources to investigate various technologies like increased body armor, new tools, and eventually new costumes and Batmobiles. The points earned from high grades on missions would either be spent buying new things on the Bat-Computer database, or getting closer to those upgrades.
Throughout, there'd be one-shots interspersed with the rest of the Bat-family. Robin, Nightwing, Catwoman, Question, Huntress, Batgirl would each get a mission or two taking place in Gotham, which ties peripherally into Batman's story, and provides him with key clues to unlock new missions. Each character would use the same system but fight slightly differently. Nightwing, for instance, would be a little quicker and stronger than Batman, but would have almost none of Batman's toys, nor the range of martial arts skills. Instead of batarangs, he'd have sticks he throws, which do substantial damage. Catwoman would have low health and strength, but would be nimble and good at hiding, with access to infrared goggles and the ability to climb up walls. Earning a high rank on their missions would unlock them on free roam mode. While these characters can't be used for Batman's main story missions, these characters could be used to deal with crime territory missions. This would also open up the possibility for the creative team to release downloadable expansions, telling individual stories for the other playable characters (because goddammit, I'd pay anything to play a Question game).