Replace part number by an acronym

The MWE in the question already contains a redefinition of \thechapter. This can be customized further by replacing the part counter \thepart with a new macro that contains the required abbreviation.

To set this using an optional argument to \part the old definition of \part can be stored in another macro such as \oldpart, and then \part can be redefined to accept an optional argument, store that in a macro, and then call the \oldpart macro which will process the default argument, similar to Optional arguments in \def. The default value of the optional argument is \thepart, to allow parts without a label to be numbered with the roman part number.

When a Table of Contents is printed the chapter labels do not fit the default width, this can be addressed using \RedeclareSectionCommand.

MWE:

\documentclass[oneside]{scrbook}
\RedeclareSectionCommand[
  tocnumwidth=1.5cm
]{chapter}

\let\oldpart\part
\renewcommand\part[1][\thepart]{\def\partacr{#1}\oldpart}
%reset chapter for each part
\makeatletter
\@addtoreset{chapter}{part}
\makeatother
\renewcommand{\thechapter}{\partacr.\arabic{chapter}}


\begin{document}

\tableofcontents

\part[ALG]{Algebra}
\chapter{Chap 1}
\chapter{Chap 2}

\part{Geometry}
\chapter{Chap 1}
\chapter{Chap 2}

\end{document}

Result:

Like this?

\documentclass[oneside]{scrbook}

%reset chapter for each part
\makeatletter
\@addtoreset{chapter}{part}
\makeatother

\newcommand{\prt}[2]{\part{#2}
\renewcommand{\thechapter}{#1\arabic{chapter}}}

\usepackage{tocloft}
\renewcommand\cftchapnumwidth{1.2cm} %<-- For tableofcontents 

\begin{document}
\tableofcontents
\prt{ALG}{Algebra} %\part[ALG]{Algebra} ?
\chapter{Chap 1}

\prt{ANA}{Analysis}
\chapter{Chap 1}

\prt{GEO}{Geometry}
\chapter{Chap 1}


\end{document}

Here is a simple solution, with an \acropart command to be used just after the \part command:

\documentclass[oneside]{scrbook}

\counterwithin*{chapter}{part}
\makeatletter
\DeclareRobustCommand\acropart[1]{\gdef\@acropart{#1}}
\renewcommand{\thechapter}{\@acropart.~\arabic{chapter}}
\makeatother

\makeatother

\begin{document}

\part{Algebra} %\part[ALG]{Algebra} ?
\acropart{ALG}
\chapter{Chapter the first}
\chapter{Chapter the second}

\part{Analysis}
\acropart{ANAL}
\chapter{Chapter the first}

\part{Geometry}
\acropart{GEOM}
\chapter{Chapter the first}

\end{document}