It’s etymologically from the notion of tapping a keg or barrel… you literally had to tap (hammer) a tap (cone-shaped cork, bung, or — eventually, but much later - a plumbed faucet or tap) into a vessel to access the contents.

Technically the video tap is the beam splitter and associated mounting affordances and hardware that redirects light bouncing off the mirror shutter to both the operator’s viewfinder and to the video assist camera; you could mount a 100% tap, replacing the viewfinder entirely, or various degrees of partial beam splitter.

In either case the tap allowed you to capture video footage of what the camera operator was seeing… which, confusingly, was exactly what did NOT actually end up in the final film, since both the viewfinder and assist can only receive light when the shutter is closed and the film is moving to the next frame.