A quick dictionary search shows the following:
1: a long arched building or gallery
: an arched covered passageway or avenue (as between shops)
: a series of arches with their columns or piers
: an amusement center having coin-operated games
The video game usage comes from the notion of a covered passageway or gallery. These galleries were often filled with mechanical, coin-operated games. (Penny arcades, etc.) Video games were the descendants of these forms of entertainment, so the name was extended to them.