It could be as simple as just 'you have detective character X, Y, or Z' (Batman, the Question, Elastic Man, J'onn J'onnz, whatever), the game offers extra dialogue choices and/or highlights 'clues' on the map that you couldn't otherwise click on.
And if you want to dodge the 'Batman gives mega-moneys, so don't put him in danger' problem... mission-end bonuses only apply to characters who go on the mission.
Well, if we're going by the comics, his main function is to clutch at his head and moan something about 'evil... so... overwhelming' before collapsing to illustrate the power and nastiness of the baddie-of-the-week, moan "Fire! My one weakness!" (or "Fire! My dread weakness!", depending) after being set on fire because the baddie-of-the-week is smart and prepared, rather than massively powerful (assuming that by 'Manhunter' you mean J'onn J'onnz, rather than Kate Spencer or any of the... was it really seven others..? that used the handle 'Manhunter').
And if we're following JLU's example, then you stuff him in the Watchtower, call him 'ops' and never let him out again until he retires and leaves the team.
... more seriously, for gameplay balance? Two simple options present themselves: one, he's bugger expensive to bring along, so if you bring him, then the rest of your team is going to consist of Snapper Carr, L-Ron and Vibe. The other is tying back into the public approval rating thing; if the Martian Manhunter is too high-profile in your missions, then people start getting nervous about the League, approval drops, you start losing permissions to operate in certain areas, funding goes down, villains start packing flamethrowers a lot more often, Lobo shows up working for the Collector (or whatever his DC equivalent is) to collect the Last Martian, whatever.
... or he could just be a bit of a Prima Donna, and when you use him too much, he wanders off to sulk ('meditate to clear his mind'... )