Why are thermionic valve/tube heaters generally designed to be 6.3 V?
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:


"A" battery for heaters. As heaters require a lot of power, this was a rechargeable battery. A 6V lead-acid battery usually is at 6.3V, so this voltage was chosen as standard.
"B" battery for anodes. This was a high voltage non-rechargeable battery, it lasted longer than the "A" battery though.
"C" battery for negative grid bias. As grids do not really use any current, this battery lasted very long.


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.