Why is the output of a filter considered 70.7% of input while determining the cutoff frequency?
The big advantage of the 50% power point is its symmetry : if you interchange the R and C in the classic low pass filter in Andy's answer, you get a high pass filter with the same cutoff frequency.

If you chose any other point (such as 50% of voltage, 25% of power) you would still get a high pass filter ... but you would have to re-calculate the cutoff frequency because it would be different for the same component values.

Only the 50% power point (-3dB gain, 3dB  attenuation, 0.707 voltage) gives you the same cutoff frequency for equivalent filters with the same components.

With hindsight it's pretty obvious that passing 50% of the power is equivalent to stopping 50% of the power, and that holds for no other ratio.

This is exploited in some filter design texts by focussing in depth on one form of filter (generally the low pass filter) and describing the other filters in outline. You can design a high pass filter by following the low pass filter design process, then following a simple process to derive the equivalent HP (or bandpass or bandstop) filter you really wanted.