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Why are the heaters generally 6.3 V (or multiples thereof)?
Does it make for an easy transformer winding ratio when used in countries with 120 VAC or 240 VAC mains?
They often pull quite significant current, and as there's generally a higher voltage supply available, a higher voltage and thinner wire could have been used.
What are the advantages of having a low voltage supply for the heaters?
When vacuum tube radios were invented, only a fraction of houses had mains electricity, therefore, first radios (and their tubes) were battery powered, they used three batteries:
I guess the 6.3V heaters continued to be used just because there was no real reason to change the voltage. Using a high voltage (220V) heater would be problematic because you would need a very thin wire for the heater (220V 9mA heater would need a really thin and long wire) and the high voltage may affect the signal in the tube.
Some tubes were designed to powered from the mains, their heaters were designed so they all draw the same current (at different voltages).
Later tubes intended for battery operation used 1.2V or 2.4V heaters which is a multiple of a NiCd battery voltage.
6.3V tubes became common at the time the first car radios (and, I suspect, other vehicle-mounted electronics, usually not for personal civilian use) were developed. 6V was the standard for car batteries back then; 12V applications were easily dealt with by intelligently building series circuits of heaters - heater currents @6V were specified in the datasheets. DC voltage converters were awkward and costly to build in these days (though often needed for the anode voltage anyway - but why make them bigger or more complicated than necessary), so designing a valve series from the ground up for car use was the most economic solution.
The voltages were selected to minimise the current from available batteries to gain the longest duration of heating.
However as the voltage increased the effects of the voltage on directly heated filaments would affect the bias point at the non grounded filament end related to the grid voltage. It would cause a spread in the bias and gain if DC heated with possible problems in reaching cut off. It would also cause a massive AC component in the cathode current that would be amplified if the filament was AC heated.
Some of this noise was removed by using centre grounded AC filaments to have the opposite ends cancel some of the problem and also indirect heating of the cathode when practical to hide the filament potentials.
A voltage of 6.3V was a compromise that took as much of the limitations into account as possible. It was close to the 2V and 1.5V cell chemistry multiples, that let one use 3 lead acid cells or 4 zinc chloride cells.
