Why is it called hyper-plane?

When we're working in a general dimension $n$, the general case we're thinking of is that $n>3$. In that case a "hyperplane" has larger dimension than the 2-dimensional planes we're used to from solid geometry.

The "plane" is the important part of the word; "hyper" just reminds us that "this is not necessarily your ordinary kind of plane from 3D geometry".

The "hyper" just refers to the dimension being in excess of the dimension we're used to, for a plane ($2$ dimensions.)

It's the same for "cube" and "hypercube." It's just a way to differentiate a general term from one that is "fossilized" in most people's minds.