Since the energy drain would be constant with no provision for energy savings by lighting only one room or a portion of one room I wonder how much savings would accrue.

I agree with Sand. Most of the time, you only need to light one or two rooms. It would waste power if the single light source was high wattage. If you reduce the wattage, the light would be too dim. What the world really needs is high-efficiency LED lightbulbs that could be used in standard incandescent sockets.

The part of this idea that uses fiber optics to direct sunlight into buildings during the day is a good one that was suggested to me years ago from a PhD in optics. The reason it is not adopted is that optical fiber is still pretty expensive, so there would be a large up-front cost to installing it. However, the benefit of millions of buildings not wasting energy on lighting during the day has non-monetary benefits that I think justify the cost. Perhaps some kind of tax incentive is required.

The part about using one central light for light at night is less efficient than using LED lighting, because the optical fibers do lose some light during transmission, and because it would be nearly impossible to capture 100% of the light from your light source with the fibers. You would also have to waste light in order to dim the lights.

Think of it this way: if your light was bright enough to light 5 rooms, and even discounting inefficiencies in transmission and capture, you could only use your system at 100% efficiency if all 5 rooms wanted maximum brightness. In this use case (discounting other losses) it would be as efficient as putting LED lighting in each of the 5 rooms. In all other cases, you would have to block light from going into some of the rooms, and then the power required to generate that light would be wasted.