How to replace a mathfrak uppercase letter by a "large" lowercase

1. Scaling is surely good with \scalebox

2. Define a \mathfrakS command along these lines

   \documentclass{article}
   \usepackage[T1]{fontenc}
   \usepackage[utopia]{mathdesign}
   \usepackage{graphicx,amsmath}
   \newcommand{\mathfrakS}{\mathpalette\bigmathfrakS\relax}
   \newcommand{\bigmathfrakS}[2]{\scalebox{1.6}{$#1\mathfrak{s}$}}
   
   \begin{document}
   \begin{gather}
   \mathfrak{s}(t) = \mathfrakS\cos(\omega t) \\
   \mathfrakS_{\mathfrakS_{\mathfrakS}}
   \end{gather}
   \end{document}
   

   

3. Use \mathfrak{S} and the students will learn it. Or use a different symbol altogether.

---

Alternative approach, which uses \mathfrak{S}, so that you can remove the macros when you decide that a Fraktur S is not a problem any more:

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utopia]{mathdesign}
\usepackage{graphicx,amsmath}

\let\ORGmathfrak\mathfrak
\def\mathfrak#1{{\ifnum\pdfstrcmp{#1}{S}=0 \mathfrakS\else\ORGmathfrak{#1}\fi}}

\newcommand{\mathfrakS}{\mathpalette\bigmathfrakS\relax}
\newcommand{\bigmathfrakS}[2]{\scalebox{1.6}{$#1\ORGmathfrak{s}$}}

\begin{document}
\begin{gather}
\mathfrak{s}(t) = \mathfrak{S}\cos(\omega t) \\
\mathfrak{S}_{\mathfrak{S}_\mathfrak{S}}
\end{gather}
\end{document}

Try using the relsize package:

\documentclass{article}
\usepackage{amssymb,relsize}

\begin{document}
\[
  \mathfrak{s} \mapsto \mathlarger{\mathfrak{s}}
\]
\end{document}

If you want yet larger font, repeat the command.