Why is \mid so called?

I can't claim any special insights into Don Knuth's linguistic preferences. I can tell you something about the use of \mid, though. The instruction \mid (in math mode, of course), produces a vertical bar with a "thick space" on either side. A good use of this symbol, for example, is when one needs to denote the fact that the material to the left of the vertical bar is being conditioned on what's to the right of the vertical bar. For instance, if X is a random variable, one might denote its expectation conditional on the r.v. being greater than 0 as $\mathrm{E}(\,X\mid X>0\,)$:

Note that this could also be produced, far less succinctly, as $\mathrm{E}(\,X\;|\; X>0\,)$ because \; inserts a "thick space".

If you want larger versions of the vertical bar as a relational symbol, while preserving the nice spacing on either side, you can write \bigm|, \Bigm|, \biggm|, and \Biggm|.

There is actually no \middle construct in the orginal TeX engine. This omission is addressed in eTeX ("Extended TeX") and successors such as pdfTeX. I.e., something like \middle| will work on most systems that are based on eTeX or one of its descendants. For more on \mid and \middle, you may want to check out the UK TeX FAQ topic Set specifications and Dirac brackets. One caveat, though: \middle, unlike \mid, will not automatically insert any space around the vertical bar; that's left to the user...

Here are two examples out of many where middle vertical bar is needed

\documentclass{article}
\usepackage{amsmath,amssymb}
\begin{document}
\[
X = \left\{x\in\mathbb{R} \middle| \int_0^xf(z)dx<0\right\}
\]
\end{document}

and for mostly inline math

\documentclass{article}
\usepackage{amsmath,amssymb}
\begin{document}
A sentence for $X = \left\{x\in\mathbb{R}\mid x<0\right\}$ some more math.
\end{document}