Objection!
Fans of the Ace Attorney games love shouting “objection!” and putting the bud guys behind bars. Daredevil happens to be a lawyer when the sun’s up. So, the only logical choice would be to lawyer up and put the bad guys away.
When he isn’t fighting the Hand, Matt would be taking cases at his firm, Nelson and Murdock. Maybe the player will cross paths someone who suffered financial losses from the Kingpin’s attempts at gentrification, and was screwed out of their money by their insurance company. These missions would have a greater emphasis on stealth than the combat oriented affairs against the Hand, requiring you to sneak around and gain information to help you in your case the following day. A failure to win the case not only decreases your reputation as a firm, it also takes away some XP from your interrogation or detective skill.
Tying these missions in with the Hand, it’d be easy for the organization in this day and age to use shell companies as a front for transporting their weapons or drugs around Hell’s Kitchen. Like Shadow of Mordor, you’d interrogate underlings for information on the lieutenants running each part of the operation – guns, drugs, transportation, etc. Taking them down and convicting them nets you a huge XP bonus. But a failure to convict leads to increased security if you try to go after them a second time.
There also should be a button dedicated to saying “lawyer up, asshole!” when handing someone over to the cops and one dedicated to saying “objection!” in the courthouse. For science, you see.