How do I recolor an image? (see images)

Easiest would be to use ColorMatrix for processing images, you will even be able to process on fly preview of desired effect - this is how many color filters are made in graphic editing applications. Here and here you can find introductions to color effects using Colormatrix in C#. By using ColorMatrix you can make colorizing filter like you want, as well as sepia, black/white, invert, range, luminosity, contrast, brightness, levels (by multi-pass) etc.

EDIT: Here is example (update - fixed color matrix to shift darker values into blue instead of previous zeroing other than blue parts - and - added 0.5f to blue because on picture above black is changed into 50% blue):

var cm = new ColorMatrix(new float[][]
{
  new float[] {1, 0, 0, 0, 0},
  new float[] {0, 1, 1, 0, 0},
  new float[] {0, 0, 1, 0, 0},
  new float[] {0, 0, 0, 1, 0},
  new float[] {0, 0, 0.5f, 0, 1}
});

var img = Image.FromFile("C:\\img.png");
var ia = new ImageAttributes();
ia.SetColorMatrix(cm);

var bmp = new Bitmap(img.Width, img.Height);
var gfx = Graphics.FromImage(bmp);
var rect = new Rectangle(0, 0, img.Width, img.Height);

gfx.DrawImage(img, rect, 0, 0, img.Width, img.Height, GraphicsUnit.Pixel, ia);

bmp.Save("C:\\processed.png", ImageFormat.Png);

The formula for calculating the new pixel is:

newColor.R = OldColor;
newColor.G = OldColor;
newColor.B = 255;

Generalizing to arbitrary colors:

I assume you want to map white to white and black to that color. So the formula is newColor = TargetColor + (White - TargetColor) * Input

newColor.R = OldColor + (1 - oldColor / 255.0) * TargetColor.R;
newColor.G = OldColor + (1 - oldColor / 255.0) * TargetColor.G;
newColor.B = OldColor + (1 - oldColor / 255.0) * TargetColor.B;

And then just iterate over the pixels of the image(byte array) and write them to a new RGB array. There are many threads on how to copy an image into a byte array and manipulate it.