What is the name of the box that the dividend goes into in a division problem?

Answer 1

There is no name for it.

You can call it the "long division symbol" though.

In terms of typography, it is actually a right parenthesis and then a vinculum (basically the up-side version of an underscore or underlining).

)_______

Notice that I produced the symbol (up-side-down) using a parenthesis and some underscores.

You can even type a vinculum using special characters: simply type in:

4 )& #x305;1& #x305;2& #x305;4& #x305;

but remove the spaces between the & and the #, and you get:

4 )̅1̅2̅4̅

A number or letter, followed by & #x305; will produce that number or letter with a line over it.

See:
http://mathworld.wolfram.com/LongDivisionSymbol.html

Answer 2

It is called the "thing," as "the thing that the dividend goes into."