At every mouse click in the mouse_click event at first you dispose the image of the picture box and then again assign the bitmap to the picturebox image otherwise you might face out of memory exception if you click on the grid too many times
private void InitializePictureBoxImage(PictureBox pictureBox)
{
if(pictureBox.Image != null)
{
var image = pictureBox.Image; // cache it to dispose
pictureBox.Image = null; // don't dispose an used object
image.Dispose(); // and dispose of it
}
Bitmap bitmap = new Bitmap(pictureBox.Width, pictureBox.Height);
pictureBox.Image = bitmap; // assign bitmap to image
}
Then call the method in the mouse_click event which you used to draw the picture box when you press the Generate Grid button in your form. Afterwards use graphics inside the mouse_click event to colour the grid which you pressed taking the X-Y coordinate from MouseEventArgs.
Altogether I think it will be something like this
private void Mouse_ClickEvent(object sender, MouseEventArgs e)
{
InitializePictureBoxImage(pictureBox);
DrawThePictureBoxImage(pictureBox);
using (Graphics graphics = Graphics.FromImage(pictureBox.Image))
{
Color color = Color.Blue;
color = Color.FromArgb(150, color.R, color.G, color.B); // lower the opacity so the writing in the grid is visible
var brush = new SolidBrush(color);
// Calculate the starting X-Y coordinate and Width,Height of the grid to color
graphics.FillRectangle(brush, startX, startY,
width, height);
}