How large does an orbiting space ship have to be in order for it to be visible from earth? I'm assuming I could crunch out some numbers to determine this, but I can see the calculations getting quite complicated. I also understand that its visibility might be effected by day light, so if it can only be discernible at night that's ok.
You can see hundreds of satellites with the naked eye.
Exact conditions for visibility will be dependent upon altitude, size, and reflectivity. As reported most of the visible ones are typically 20 feet or larger across.
With just the right angle, solar panels could be considerably smaller as they are very flat surfaces that reflect some light like a mirror.
Sputnik-1 was barely visible to the naked eye, magnitude 6. A sphere 23 inches in diameter and highly polished. This is about as small as possible, unless you consider something smaller that focuses the reflected sunlight to the observer (like a solar panel).
An active light emitter such as a small cluster of LED bulbs could be smaller yet and be visible as long as the total lumens would result in a 6th magnitude source.
Perhaps you actually intended to consider the resolving power of the human eye instead. What it the difference? Thousands of stars are visible, but only the sun has a visible diameter. I.e. of all the stars, only the sun can be resolved by the naked eye.
Generally, 20/20 vision is considered to have a resolution of about 1 arc-minute, which would require something about the size of the ISS (100 meters) when under optimal viewing conditions.