What is the role of Tokyo in the heist?

(Note: I have only seen Season 1 till now)

Well, I would say that Tokyo is a key character in the group. At first sight she doesn't have any special skills, like Rio does in technology stuff, or Moscow in safes.

But, let's consider these facts:

• She has a criminal background

  This is a "robbery" and you cannot go inside the Royal Mint with a group which only consists in people with some technical expertise in different fields. You are about to commit a crime, and you a need a felon who has experience on these things to increase the probability to get out without being captured.

• She has a good control of emotions (still far from Berlin though, he is almost a machine)

  The plan was to stay inside the building for 12 days. It's a long time and everything can happen: police trying to enter the building, hostages will try to escape or to harm you, conflicts within the group etc. There is a lot of pressure. You have to have cold blood people to handle that pressure. You need people who don't scream when they see blood or a dead body, people who don't mind to put the gun to someone's head without shaking the hand and are even ready to shoot if necessary (Rio doesn't fall in here). You need people who can take decisions under pressure and who can take control of the situation when it escalates.

  Remember the scene after 16 hostages escaped, when the teacher started to applaud, and other hostages followed her... What did Rio do? He was about to cry, he couldn't handle the pressure! WTF? How could this happen? He is the grabber, they are the victims. He has a gun in his hand, he is the "felon", the bad guy. He is supposed to be the one who has nothing to lose and doesn't fear anything. The hostages should fear and obey him, not the opposite. Where is the reputation? That was intolerable.

  This is exactly the moment when you need someone like Tokyo or Berlin. You need people who can take the psychological superiority in any time. You need people who are able to show in any time who is the victim and who is the persecutor.

• She won't step back for any reason

  The Professor said that everyone in the group has nothing to lose, but I'm not sure if all members of the group were conscientious about this. Rio had given a prove, when he was thinking to surrender after he saw the video message from his parents. He was considering to give up and this goes against the "nothing to lose". Instead, Tokyo was one of them who never considered the surrender as an option. She knew perfectly and never doubt it what her goal was, and why did they started this adventure. She would be out of the building either dead or with the money. NO third option here!

So, to conclude and answer your question, to me Tokyo's role is to get things done when other teammates cannot (because they are weak or for any other reason).

People/characters like this are decisive in solving conflicts and problems, and in general to take control over things in tension situations.

EDIT (after the 1st comment)

That's true, she was the first one to break that rule. The purpose of the rule was to prevent emotions to affect decisions related to the final goal. More than emotionally, she was physically engaged with Rio. She was a free spirit, and she had sex either for pleasure or to relieve stress.

The "NO RELATIONALSHIPS RULE", comes to me as "NO EMOTIONALLY INVOLVED WITH SOMEONE". As long as she can keep the distance between physical contact and emotional contact, there is no problem.

I think she wasn't emotionally involved with Rio because:

1. She isn't the type that falls in that easy.
2. Rio is not her type.
3. She flirted with Denver while she was in a "relationship" with Rio.

Even though she started that relationship, considering her temperament, I believe that if she would have to choose between the money and Rio, she would choose the former option.

Season 4 went a bit more in depth with Tokyo's reason for joinig the heist. Paraphrasing Tokyo's conversation with the Professor: when things get dangerous, Tokyo acts instead of freezing. She excels at this compared to the others.

This already happens in season 1, when she ran through gunfire to get the mounted machinegun and essentially save the heist as SWAT had almost breached the building.