I know there’s the generic term, “rectangular prism”, but is there a one-word term for it (like sphere, cube, etc.)?
Box.
hexehedron?
Cuboid is a more technical-sounding term, but ‘box’ really is pretty common.
For another fancy term, a box is a case of a parallelepiped (think of a parallelogram extended to three dimensions).
Seems like a cube is a 3D square by definition. Looks like a 3D rectangle, in general, is a prism.
http://mathforum.org/workshops/sum98/participants/sanders/Geom3D.html
In school, I remember calling the form a rectangular prism.
A right hexahedron.
Put me down for rectangular prism as well.
Damn. Cuboid.
From here
The “official” term is rectangular parallelopiped, but box is what I call it.
But there’s no requirement for the “sides” of the box to be perpendicular to the top and bottom in a parallelopiped is there?
Not in general, but if all the faces are rectangular, it has to happen like that.
rectangular parallelopiped - ahh … sweet memories of Calc III
No body else would call it a pyramid?
Well then what would you call a pyramid?
It’s definitely cuboid. Much more concise than parallelopiped, and more specific, too.
Box is twice as concise as cuboid. 
Box is the first thing that comes to my mind.
Boxes have lids.
“Box” is just silly. A box is a physical object, which may or may not be cuboid in shape (eg think of a hat box, which is cylindrical). “Box” is certainly not a mathematical term, unless in some kind of horribly dumbed-down school textbook that doesn’t want to scare the little darlings with proper words.
Ahem, rant over.
Anyway, the mathematical term for “a geometric solid whose six faces are rectangles” (definition from Collins English Dictionary) is “cuboid”. It is a special case of the parallelepiped - also called a rectangular parallelepiped (note spelling, which I got wrong too in my last post!)