I just had this curious question. In other fields, the word "hyper" is actually used to refer to something which is "over; beyond; above" as defined by Google. An example of such terms would be Hyperglycaemia, defined by Wikipedia as:
a condition in which an excessive amount of glucose circulates in the blood plasma.
Or Hypersonic, defined by Google as:
relating to speeds of more than five times the speed of sound.
So, my question is, what would be the motive to give a Hyperplane such a name when it is defined as :
a subspace of one dimension less than its ambient space
Adding hyper- helps the listener know that it is like a plane, and yet it might be more than just an ordinary 2-d plane.
From 3-d geometry, we think of planes as being 2 dimensional things in a 3 dimensional world. In regular Euclidean geometry, there is a unique plane normal to a line through a given point. Similarly, there is a unique line through a given point on a plane that's normal to the plane. We want to take this line-plane connection through to higher dimensions. That's why the definition of "hyperplane" says it is $n-1$ dimensional, because it is the orthogonal complement to a $1$ dimensional piece of the space.