A street sign has two magnetic ballasts, 120 volts, total load of 8 amps. The run is 250 feet. When I do the math with #12 copper I get 112 volts at the load. Will low voltage affect ballast operation or ballast life?
Provided your math is correct, (I did not calculate it), the output from the ballast will be relative to the input. The lamps will operate sufficiently. The ballast is a XFMR. What goes in, is reflected by what goes out. At lower voltages the lamps lose brightness. Since this is an outside application, temperature plays a major role in the brightness of the lamps. I always recommend high output ballasts for outside use. At the lower voltage it may take some time for the lamps to achieve effective lighting. Even high output ballast's struggle in cold weather. If possible, leave the lights on 24-7. I would run #10 copper.
Have you considered a 12ga multi-wire branch circuit?
This would give a 3V drop if the supply is 208/120, and less than 2V if the supply is 120/240. This would be better performance than a 120V supply made with #10 conductors, and would have less total copper.
Rather than running a 120V circuit consisting of a single ungrounded (hot) conductor and a single grounded (neutral) conductor, I am suggesting running two 120V circuits with a shared neutral, consisting of two ungrounded conductors from two different supply legs and a single grounded conductor. This is called a multi-wire branch circuit (MWBC), the benefits and issues of which are regularly debated. I am not saying that this _is_ the best circuit to use in this case, simply suggesting that it be considered.
In the ideal 'balanced' case, no current flows on the neutral, so that there is no voltage drop on the neutral. In the case of two legs from a three phase system, or in the case of harmonics, there will be voltage drop on the neutral, but it will be reduced from the single circuit case.
Rather than 8A on a single circuit, you have 4A on each of two circuits.
The net result is that the length of conductor over which you consider voltage drop is reduced, and the current flow causing the voltage drop is reduced. The benefit of the MWBC is so large in this case that the voltage drop is lower if you use 12ga conductors in this configuration than if you use 10ga conductors in a single circuit configuration.
4 #12 conductors contain less total copper than 3 #10 conductors. (Remember the EGC!) So the wire is probably less expensive even though you have more conductors.
The cost adder is that you need a double pole breaker, and double pole switches.