I think an investigation mini game is probably more important than a social mini game. Honestly, we wring our hands about a social mini game, and how it can break down over people doing things they might realistically do in an actual social "contest" (the nuke in the briefcase thing), but, honestly, I think... you should just roll with that, and let people socially interact in game they do irl, and let the dice and skills be there to fall back to if players or the GM want to use them, and to determine whether your character can convince another character of something that isn't pretty reasonable (let me go in this locked and guarded room, as opposed to "hey, can you watch my briefcase while I go piss? Don't worry about what's in it")
But investigation doesn't really work the same way. At least outside of grilling suspects, I suppose. You need to know whether the PC can find a hidden thing, how well it's hidden, whether they can recognize that it's a clue, whether they can make connections, and so on.
Harshax mentioned Gumshoe, and I have looked at it a bit. I ultimately decided I didn't want to use it when I was working on an investigation game, but I do think it has some good ideas. IIRC, the "you can use multiple skills to search for things relevant to those skills" and "you always find something| come from Gumshoe.
Last year, I decided I wanted to run a game set in the ALIEN universe as a Halloween one-shot, and decided I wanted it to be pretty investigation focused. I wasn't really ready to go all in on the actual ALIEN RPG, especially since I would have to learn it to write something up, and then teach it to my players. Instead, I used FATE, which I think is a very good system for one-shots. And I know it, at least well enough. You can pretty easily write up an adventure and a set of pregen characters if you want to do that, and so on in FATE. And the fact that it's pretty light means that it can be easily taught to others.
But I did have to figure out how to make an investigation happen in FATE. As well as how to write a mystery adventure at all. I did draw on the idea mentioned above with Gumshoe, where you can use different skills, and you always find something. I would have handled looking for clues by just saying "tell me what skill you're using to investigate. Say the right skill, you get at least this much, and some clues you can roll to get more." Investigation that involved talking to people was definitely going to care about rolls, though. One tip I found when googling how to write a mystery for an RPG was to have three clues at each scene where players investigate, and those clues should lead to the other scenes.
That's all I really did when I wrote an investigation adventure, though I can't say how well it worked since the game didn't happen. But I think those are three important parts of an investigation mini-game:
- People can use multiple skills to look for clues--and if you're making pregen characters, make sure each character has at least one investigation skill. If you're not Pregen-ing characters, the players should be guided to have at least one specific Investigation skill, and if you're writing an Investigation Heartbreaker, you can do this by having a list of such skills and a character gets n, where n is at least 1.
- Corollary, you should probably not have a skill called Investigate. If a clue is found simply by saying "I snoop around the physical location," maybe it should just be Notice rather than Investigate.
- When investigating, a PC always gets something, and you should probably do this by saying "your job is to investigate. As long as you are in the right place and say you're investigating, you will get at least one clue, with no dice/DC involved."
- Each investigation scene should have at least three clues specifically called out, and at least two of those clues should lead to other scenes for investigation.
Each archetype other than The Driver had a +3 skill with a clue in at least three of the five scenes where you had to specifically say "I'm using X to investigate" (there was also a scene where you could just talk openly with someone who had no reason to lie, obfuscate or misdirect, and that needed nothing more than asking him about some of the clues you previously found), as well as at least two other clue-relevant skills between the +2 and +1 ranks. The Courier had Comtech at +1, but being an android, they had a special ability that basically upgraded it to +3 if they used their integrated datalink, as well as also having +3 Rapport, useful for the couple of interview investigation scenes where you did actually need to get info out of someone who had things to hide.
The Driver, however, only had two investigation skills (Notice and Provoke), both in their +1 track, so they would have been of somewhat less use in investigation. I think the idea was that they would get the spotlight when combat happened, which it was going to, and would have involved a xenomorph, but, still, I should have specifically planned for the Driver to be more involved in investigation.
Thinking forward, more, if I were writing an Investigation focused game, I think I would also try to make clues more relevant than just in solving the puzzle. Maybe having found a clue could give you bonuses to other stuff in the game, like, as an example off the top of my head with the above game, if you found the picture on the victim's desk of his daughter, you could invoke that when talking to the estranged daughter to put her off her guard and get a bonus in the interview.