How best to change the font size (etc) of threeparttable's table notes?

The right command to act on is indeed \TPTnoteSettings, but its standard definition must be preserved, because it contains essential information for typesetting the notes. The best is to add to it and etoolbox provides a simple way.

\documentclass{article}
\usepackage{threeparttable,booktabs}
\usepackage{etoolbox}
\appto\TPTnoteSettings{\footnotesize}

\begin{document}
\begin{table}
\begin{threeparttable}[b]
\caption{A caption}
\begin{tabular}{llll}
\toprule
42\tnote{1} & some & text & to have room\\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item [1] the first note
\end{tablenotes}
\end{threeparttable}
\end{table}
\end{document}

I used a simpler approach which was adequate for my needs

...
\begin{tablenotes}
\footnotesize
\item [1] See \url{https://texblog.org/2012/08/29/changing-the-font-size-in-latex/}
\end{tablenotes}
...

Which was inspired by https://texblog.org/2012/08/29/changing-the-font-size-in-latex/

FYI: I added the URL in the item note as my table has a series of long URLs I needed to fit within the column width.