White balancing means making the neutral color of the scene (like a grey card) have a particular color temperature and tint in the output picture — to simulate color adaptation of human vision to the conditions of lighting in the scene.
If we take a hyperspectral photo of a grey card in a room lit by incandescent light bulbs, spectral radiance of the directions corresponding to this card will have a color temperature of about 2700 K. When doing white balancing, we normally adjust the colors in such a way that neutral colors appear as neutral on screen.
Normally, if a monitor is calibrated to a particular white point (which means its
#ffffff color (the neutral white color) has chromaticity coordinates of this white point), it's assumed that the room this monitor is in is lit by an illuminant with chromaticity of this white point. This way a grey card put side-to-side with the monitor will have the same chromaticity as the white color on the screen, which enables soft proofing (when monitor brightness is adjusted to show white as the same color as a white card next to the monitor).
Now, if we want our monitor to display (as close as possible) a scene as if we are looking into it through a window, i.e. without doing scene-specific white balancing to simulate color adaptation, the settings for color balance in the camera should be such that a grey card lit by illuminant with color temperature T had the same color temperature on output. In particular, a grey card lit by illuminant with the monitor's white point (e.g. the monitor itself*) should be displayed with color proportional to
In particular, the above means that, if you shoot a grey card lit by your calibrated monitor on which you intend to display the scene, color balancing should be done in such a way, that mean color of this card's pixels in the photo on the screen becomes neutral (proportional to
When choosing color balance settings in the camera, you should choose the one closest to the white point of target monitor (e.g. "Daylight" for 6500 K-calibrated monitor). Some cameras have a "Custom White Balance" option, which lets you take a photo of a white object and use this color as the illuminant. In this case you can use the target monitor as the object.
*The monitor colors should have constant chromaticities when viewed from different angles. Many consumer monitors don't have this property even approximately, and their "white" could e.g. look blue when looked from above and red when looked from below. These aren't useful for color-correct rendering.