OK there's a lot here so I'll take a little time to answering all your questions.
So our hero has power to influence others. Is this some kind of hypnosis or just charisma?
It's a power either based on pheromones or subconsious mind control. Functionally the same at this time so I've not decided yet exactly what it is.
Wait a second, there's another people with similar powers? how do player know about them? and how do player prevent them from getting hero's power?
There may be others with similar powers, yes. And the player knows nothing about them at the start of the game. One of the challenges is to expand the organization carefully so that you don't run into opposition that you can't handle. The player has to prevent others using powers against them by keeping below the radar until they can strike against potential threats.
Wait a second, there's battle too?
It's a possibility, but I meant fights in the sense of situations you have to handle and can potentially loose the game to if you go in unprepared.
I assume there will be meter or something to show how hidden player's base/organization is. The question is what will happen if government figures out about player's resistance?
Yes, there will most probably be a meter of awareness or equivalent for every opposing organization. The most important and impactful of them will be the one of the government.
And also other organizations? what are they?
There are probably other budding or established resistance organizations out there that can either be allied by the player or that the player will have to deal with in some way.
Also directing research. Eh? what's this and what for?
At the start the player will only possess a power that is uncontrollable and passive. To be able to reliably influence and recruit others he'll need to put effort into honing his powers. He'll also need to find ways of gathering information about potential targets and recruits (such as analyzing genetic material to find out if the target has powers and what they are). Maybe additional ways of recruiting others, and so on. I'm referring to this as research. There'll be a research tree of sorts and the player will have to set aside assets to go through this tree.
I'm lost.... how many kind of power this game would have actually?
Potentially a lot. Everything from simple ability enhancing powers (super intelligence, super strength and so on) of varying strength to invisibility, flying and so on. Think marvel X-men, Heroes and any super hero franchise. All these powers are a possibility. They just have to be written into the story and made to fit mechanically.
I'm a huge fan of moddability and I've thought about ways to set up the game in a way where the game is generated by reading through a bunch of files at startup so that people can actually create and set up their own NPC:s with powers, stats, encounters, graphics and so on. That's probably a project for much later, but I'll try to keep it in mind from the start so that I can ease the implementation when the time comes.
Although I'm not sure of the powers in this game, I could imagine power stacking suggested there. Still, when you say resetting time, what actually happens?
One of the NPC:s you meet at the start of the game is a mutant with the power of resetting time with his memories intact, including the players influence over him (think groundhog day but without the reset always happening at the same time every day, instead happening when he wills it or on death). After having reset uncountable times, this mutant is properly devoted to the well being of the player and he'll practically function as an early game help. His goal is to escape the hell he's stuck with after hundreds of millions of resets and he's found a way to pass the power to the player. The functionality of this power is that the player can will a reset at any time to the point where he got the power and time will also reset at death. Basically, a reset (or death) will trigger a game over with the option of starting anew (passed the intro) with some benefits (the nature of these are not yet decided) that'll stack with resets (how much depends on how far into the game we were at reset). At least the plan is to implement this eventually.
Thank you for your feedback and the opportunity to elaborate :)