Radar terminology - Specific meaning of "video signal"
You also see this on Spectrum Analyzers; "video bandwidth" for the same reason: the video is the output voltage from the power detector used to drive the vertical on a display (when the horizontal is the sweep), or in the case of the polar displays used to drive the radius where the sweep drives the angle. Since it is used for the display signal it is properly called the "video" signal. As I introduced, this is the same on a spectrum analyzer where we can adjust both "resolution bandwidth" and "video bandwidth": Resolution Bandwidth is the bandwidth of the effective bandpass filter prior to the power detector, while Video Bandwidth is the bandwidth of the effective low pass filter after the power detector. The first filter will reduce the total power incident on the detector when the input is spread as a spectral density, thus reducing the voltage out of the detector, while the second filter (video filter) will only smooth (average) the measured noise.

Here is a perfect picture showing what I describe in terms of the Spectrum Analyzer, credit Erik Diez of Agilent (now Keysight). 
http://electronicdesign.com/test-amp-measurement/fundamentals-spectrum-analysis



So for the case of radar, at least the original analog systems, the video output was a similar power detector used to drive one axis (either magnitude or as was common radius on a polar plot for a rotating system) of the radar output display.



(image credit: http://www.linuxwolfpack.com/swapshop.php)