This is exactly what the HSB colour model is for. You have almost exactly named the model's variables in your question:
- Hue is the 'kind' of colour: red, blue, orange, yellow;
- Saturation is inverse with the amount of white you add to the hue;
- Brightness is inverse with the amount of black you add.
So, to take your example colour of #F2F5F7. you can convert this into HSB in Photoshop or Illustrator's colour picker, yielding H=203, S=1, B=96 as HSB values.
All you have to do now is change the H to the hue you want. For a green, try H=83 and leave the S and B values intact. This is converted by the colour picker right away and yields #f3f4f2.