The first problem we have to solve is power: where does the light energy come from? Our options are the same as for every other spacecraft:
Nuclear thermal: Although this option provides a lot of power, there are two strikes against it: nuclear reactors are very heavy, and they are currently not ready for spaceflight. You'd be looking at a decade-long development program, which does not meet requirements.
Radioisotope thermoelectric generators (RTG): These are a common choice for deep-space probes since they provide long-term power with high reliability. New Horizons and the Curiosity rover both use plutonium RTGs. However, they are not efficient (typical powers are around a hundred watts), and they are very expensive. (NASA is paying the US Department of Energy over a hundred million dollars to restart production at the level of about a kilogram per year.)
Fuel cells: These can provide high power by consuming a fuel and oxidizer. These consumables limit the lifespan of fuel cells, so they do not meet requirements. (I'm making the assumption that you want the spacecraft to remain visible for at least a month.)
Solar power: Solar photovoltaic is an OK option. However, we can also use the incoming sunlight directly by reflecting it at the Earth. This eliminates the need to collect, process, and re-emit all of the light power, replacing that equipment with a simple reflective surface.
The idea of beaming power from a 'generator' satellite in high orbit to a 'lightbulb' satellite in low orbit is a good one which draws on technically plausible concepts. However there are a couple reasons that I'm avoiding it in my design:
- Microwave power transmission is nowhere near mature, and would require a development period of many years.
- There is a lot of inefficiency. Even assuming very high efficiency solar panels (30%), LED lights (40%), power transmission (90%) and conversion (95%), the overall efficiency is only about 10% (and the transmission efficiency is likely to be far lower), compared to around 95% for a reflector.
- Solar panels are far heavier per area than reflectors (on the order of 100 watts per kilogram). Add to that the cost of launching that weight into high orbit and the project quickly becomes infeasible.
Choosing the passive reflector, our initial design looks something like a solar sail.
The next problem is the structure of the spacecraft. Unfortunately the reflector can't be tensioned by centrifugal force like a solar sail, because the rotation would interfere with the pointing (since the reflector is not an isotropic radiator, steerability is a requirement). I envision a folding truss structure like SMAP's antenna (Note that SMAP's budget was around 900 million dollars). The structure doesn't need to be as "dense" since the flatness of the reflector is not as critical.
Now we should determine the reflector size. Assuming that the reflector is very close to flat, it will appear (to an observer within the reflected beam of sunlight) to have the same surface brightness as the Sun. (To put it another way, it looks like a window showing a small part of the Sun.) Thus, the total apparent brightness is equal to the apparent brightness of the Sun times the reflector's apparent size relative to the Sun. To get some rough numbers, I'll assume the reflector has $90\%$ efficiency. The apparent magnitude of the reflector is:
$$
m_\text{sc} = m_\text{Sun} - 5\log_{10}\left(\frac{d/r}{32'}\right)
$$
The quantity in the logarithm is the angular size of the spacecraft (it's diameter $d$ divided by distance to the observer $r$) divided by the angular size of the Sun (in minutes of arc). The apparent magnitude of the Sun is $-26.74$. Putting this into a plot, we get:

We can see that this rough estimate has good agreement with the magnitude of Iridium flares, caused by reflective antennas 1–2 meters in size.
Assuming again that the reflector is close to flat, the width of the reflected beam will be around 30-40 arcminutes. The diameter of the beam at the surface will be about one percent of the reflector's altitude (from a 4 km spot in low orbit to a 350 km spot in geosynchronous orbit).
However, we need to maximize not the peak magnitude, but the average magnitude. This is affected both by visibility of the spacecraft from the ground and shadowing of the spacecraft by the Earth.
I performed some simulations to determine the optimal altitude. I took into account four factors:
- Visibility: If the spacecraft is below the horizon, it is not visible to the observer and its relative brightness is $0$.
- Viewing angle: Imagine extending two lines from the spacecraft, one to the Sun and one to the observer. Call the angle between these lines $\theta$. If $\theta=0$, the Sun is directly behind the observer as they look at the spacecraft and the relative brightness is $1$. If $\theta=\pi/2$, the reflector must be turned at 45 degrees and its apparent size (and apparent brightness) is only $1/\sqrt{2}\approx 71\%$. If $\theta=\pi$ the spacecraft is directly in between the Sun and the observer; in this configuration the spacecraft can't reflect any light towards the observer and its relative brightness is $0$. The relative brightness is $\cos(\theta/2)$.
- Distance: The apparent brightness of the spacecraft is proportional to the inverse square of the distance to the observer.
- Shadowing: I assumed that the Earth's shadow is perfectly sharp, so the spacecraft is either fully illuminated (brightness $1$) or fully shadowed (brightness $0$).
Surprisingly, the simulations indicate that you want the reflector to be in as low an orbit as possible. The decrease in brightness with altitude dominates the increased visibility. However, our orbit can't be too low, since with a large, lightweight structure drag becomes an issue.
Let's spec a 1000 kilometer orbit. At this altitude, even a 100-meter reflector would drop by only about ten kilometers per year. The inclination should be a little higher than the latitude of the northernmost (or southernmost) location you want it to be visible from. The reflector will cast a spot about ten kilometers wide on the Earth's surface.
At 1000 kilometers, we could make the spacecraft as bright as the full moon with a 20-meter reflector. This size is easily within the realm of plausibility; JWST's sunshield is close to this size at 18 meters long.