How to select an area on a PictureBox.Image with mouse in C#

I used your code, you were nearly there. You needed to Invalidate the pictureBox1 instead of the rectangle. I also added a check for the Rect so it doesn't get drawn when it's not initialized or has no size.

Another important change: I created the Rectangle only once and I adjusted its location and size. Less garbage to clean up!

EDIT

I added a mouse right-click handler for the Rectangle.

private Point RectStartPoint;
private Rectangle Rect = new Rectangle();
private Brush selectionBrush = new SolidBrush(Color.FromArgb(128, 72, 145, 220));

// Start Rectangle
//
private void pictureBox1_MouseDown(object sender, System.Windows.Forms.MouseEventArgs e)
{
    // Determine the initial rectangle coordinates...
    RectStartPoint = e.Location;
    Invalidate();
}

// Draw Rectangle
//
private void pictureBox1_MouseMove(object sender, System.Windows.Forms.MouseEventArgs e)
{
    if (e.Button != MouseButtons.Left)
        return;
    Point tempEndPoint = e.Location;
    Rect.Location = new Point(
        Math.Min(RectStartPoint.X, tempEndPoint.X),
        Math.Min(RectStartPoint.Y, tempEndPoint.Y));
    Rect.Size = new Size(
        Math.Abs(RectStartPoint.X - tempEndPoint.X),
        Math.Abs(RectStartPoint.Y - tempEndPoint.Y));
    pictureBox1.Invalidate();
}

// Draw Area
//
private void pictureBox1_Paint(object sender, System.Windows.Forms.PaintEventArgs e)
{
    // Draw the rectangle...
    if (pictureBox1.Image != null)
    {
        if (Rect != null && Rect.Width > 0 && Rect.Height > 0)
        {
            e.Graphics.FillRectangle(selectionBrush, Rect);
        }
    }
}

private void pictureBox1_MouseUp(object sender, MouseEventArgs e)
{
    if (e.Button == MouseButtons.Right)
    {
        if (Rect.Contains(e.Location))
        {
            Debug.WriteLine("Right click");
        }
    }
}