player controls menu

Depending on the design of your game, you can put the key binding information in a specific class or as a static class variable.

 public class GameKey {
    public static KeyCode Left = KeyCode.A;
    public static KeyCode Right = KeyCode.D;
    ...
 }

To use them:

 if( Input.GetKey( GameKey.Left ) ) {
    //Move your character left
 }

To do the binding, there are different ways to do it.

1. Standard form: a label and a textfield, update the key binding whenever there are changes.

2. Press a key: a set of buttons, when a button is press, prompt the user for input, the input is set as the key.

3. Other ways that I did not think of as of this writing

So I will leave that to you to figure it out.

 void OnGUI() {
    //Standard form
    inputString = GUILayout.TextField (...);
    GameKey.Left = Enum.Parse(typeof(KeyCode), inputString ); //Convert string to enum

    //Press a key form
    .... //Some kind of prompt
    GameKey.Left = e.KeyCode; //e is the Event.Current;
 }

You will also need to save the key binding and load them (this also means that you will load the GameKey data using a function on the game start). This is a topic I am not familiar with, but I do think that it is doable using PlayerPrefs.