Investigation games are tricky beasts to run with a single player. For starters, the investigation genre requires a lot of brainpower and mini epiphanies to make it from the beginning to the end. While having, say, four or more players tends to increase the number of tangents, misunderstandings, and arguments about the 'truth' exponentially with each player added, having only one player means that it's all on them. In truth, I think two to three players is the sweet spot for investigation as it means there are multiple perspectives without having so many that part of the game has to be spent on consensus building on what a clue is let alone what it means.
So what do you do if you have a solo player?
Well, I'd recommend giving them a bit more time to think than you normally would if you had multiple players. Rather than demanding that they come up with the answer within the confines of the character's amount of time (matching in-game time to out-of-game time), I would recommend giving them more time and perhaps a few nudges to go over old clues and timelines if necessary. The goal here is to ensure that they do have that epiphany (don't rob them of that moment) but the process shouldn't lead to hair pulling and frustration.
Tension is good. Frustration is bad.
Basically the rule with players no matter the number.
So what if you have included an NPC buddy or so to help them? Perhaps you included them for social reasons because investigation games can get a bit lonely otherwise. Or perhaps you wanted to shore them up in a few of their character's weak points. Or simply wanted a Sherlock Holmes and Watson vibe. Or perhaps the player latched onto an NPC and dragged them along with them. Or it's simply logical that the person would have a partner.
Whatever the case, what do you do now?
Well, the NPC must never steal those plot defining epiphanies. That sense of satisfaction belongs to the player, not to a character run by someone who knows all of the answers. However, the NPC could make the odd connection, highlight the importance of a prop, remember that a name has been mentioned a few times before, or even suggest a few places to go next.
The limitation on this is that the NPC must not solve the plot nor connect too many dots. Their role is to support the protagonist, after all. Sure, they can have the odd moment in the limelight to keep them interesting but have that occur with the occasional brilliant moment of good cop / bad copping, or have them have minor epiphanies about the significance of something the player character forgot. Or even a red herring. Perhaps they are fantastic in a tangential situation, shedding light on the relationship between a witness and his secretary. Does it matter to the plot at large? No. But it could show the NPC's insight into social interactions without stealing any of the limelight from the player.
The NPC certainly can make rolls to perform autopsies, perceive a clue, break and enter, and anything else that the player character can't do or has already failed to do.
What about when interviewing witnesses or talking to people?
The NPC can ask the occasional question, provide a clarifying comment, or perhaps be the good cop to the PC's bad cop (or vice versa) but they should very much play second fiddle here. Not only is it difficult to hold a conversation with yourself using two personalities as well as a second actual person, but it's also boring to see and watch such a thing for too long.
And with the occasional combat? Just make sure your NPC's combat statistics aren't as great as the PC's and always roll the dice rather than letting your NPC automatically achieve their efforts. Unless, of course, the player doesn't care much for combat and has explicitly made a character that's largely a noncombatant. In that case, don't worry so much about the combats the NPC is fighting. Focus on what the PC does. Perhaps the PC is trying to escape while the NPC provides covering fire. As the PC can't see what's happening, don't describe that. Just roll a few dice (whether you pay attention to it or not) and just provide the occasional sound description while focusing on what the PC is doing in those rounds.
Also remember that the NPC should fail on occasion. Their advice should sometimes be incorrect. Their powers should sometimes fail. They should sometimes chase red herrings, neglect clues, and otherwise be blindsided by their own personality, their own assumptions, and their own history. This will make them more interesting as a character and ensure that the player doesn't quickly find that the best course of action is to agree with the NPC.
What about having the NPC do things while the PC does other things? Sure. Your best bet is to generally get them to do the boring stuff. Either something that is repetitive or which you or the player might not enjoy roleplaying so much. Anything that can be done off-screen, in other words.
Occasionally they might take the more interesting option but this should only be if it will complicate the plot and throw the spotlight all the more forcefully onto the PC. For example, you know that out of the two options, going to the old folk's home and going to the police station, that the old folk's home is the more interesting one. The NPC elects to go to the old folk's home and the PC agrees. While there, the NPC finds something critical to the case and is kidnapped for his efforts. Now the PC must track him down and free him from the bad guys. See? A far more interesting option that talking to the police but in the end the kidnapping puts more of a focus onto the rescuer. Of course, if the PC really wanted to go to the old folk's home then they should be able to do so. You can always have the NPC kidnapped in some later circumstance.
Finally, take your cues from your players and keep an honest and open dialogue with them over the course of the campaign. They should feel like they can tell you if an NPC is becoming too dominant or stole any of their thunder. You also shouldn't be upset if they do. They might love what you're doing with the NPC and want to keep him or her around. They just want more of the focus in a few particular areas. Or perhaps they're sensitive over how you handled a particular scene because of a previous Storyteller's actions or even their own neuroses. When in doubt, ask.