what to replace \it with in Math mode to make scrbook happy?

Use the option the KOMA-bundle provides to get the old font commands back. The option was considered deprecated when implemented, hence you are still getting a warning for using it (once).

\documentclass[enabledeprecatedfontcommands]{scrartcl}
\usepackage{tex4ht}
\begin{document}
\it Wombat
\bf Capybara
\end{document}

It should be mentioned though, that the font commands are deprecated for two decades now.

As you may know, old-style font commands like \rm and \it are deprecated in LaTeX. The answer at Does it matter if I use \textit or \it, \bfseries or \bf, etc explains some of the pitfalls of the old-font commands. The more modern approach uses the notions of font family, series, and shape as three orthogonal vectors of a font's definition.

Examples of font families include roman, sans-serif, and teletype. On the other hand, examples of font shape include upright, italic, slanted, and small-caps. Examples of font series include bold, medium, and light.

Thus, in the answer cited by the OP, there is already the answer to this question listed in that question. The line:

\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}

While I have no prior experience with these commands, it seems likely that it is a way of telling LaTeX to replace \it with \normalfont\itshape in text mode and to associate it with \mathit in math mode.

Thus, the addition to your MWE would be:

\documentclass[12pt]{scrbook}% 
\usepackage[T1]{fontenc}

 %from questions/57109/what-exactly-does-declareoldfontcommand-and-declarerobustcommand-do
\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}

\begin{document}
$\rm e^t  + \it p1$   %what about \it?

\[
\left[ \begin {array}{c} {\frac {\rm d}{{\rm d}t}}{\it p1} \left( t
 \right) \\ \noalign{\medskip}{\frac {\rm d}{{\rm d}t}}{\it p2}
 \left( t \right) \\ \noalign{\medskip}{\frac {\rm d}{{\rm d}t}}{\it 
p3} \left( t \right) \end {array} \right] = \left[ \begin {array}{c} 0
\\ \noalign{\medskip}-{\it q3} \left( t \right) \\ \noalign{\medskip}{
\it q2} \left( t \right) \end {array} \right]
\]
\end{document}

More can be learned at https://en.wikibooks.org/wiki/LaTeX/Fonts to see what all the different possibilities for family, series, and shape are typically supported by a font.

This is merely a suggestion for a different input. bmatrix instead of array, no \left, \right, a macro for the differential d.

\documentclass[12pt]{scrbook}% 
\usepackage[T1]{fontenc}
\usepackage{amsmath}

\newcommand\diff{\mathop{}\!\mathrm{d}}

\begin{document}
$\mathrm{e}^t  + p_1$

\[
\renewcommand*{\arraystretch}{1.5}
\begin{bmatrix}
  \frac{\diff}{\diff t}{p_1} (t) \\ 
  \frac{\diff}{\diff t}{p_2} (t) \\
  \frac{\diff}{\diff t}{p_3} (t)  
\end{bmatrix}
=
\begin{bmatrix}
  0  \\ 
 -q_3 (t) \\ 
  q_2 (t)
\end{bmatrix}
\]
\end{document}