It looks like I have found a bug using the external library in lyx:
if you pass the option
\tikzexternalize[prefix=some_dir/]
then lyx is not able to create the subdir "some_dir" in its temporary
working dir under the /tmp tree, which results in a compilation error since
lyx is not able to write the necessary files.

If you don't pass any prefix option, the file compiles correctly, since lyx
can write all necessary files in the relative temp dir.

A MWE is the attached file, which results in the following latex source.

M

---------------------------------------------------------------------------------------------------------------------------------
%% LyX 2.3.6 created this file.  For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}

\makeatletter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\usepackage{tikz}
\usetikzlibrary{external}
\tikzexternalize[prefix=picts/]

\makeatother

\begin{document}
\[
 \begin{tikzpicture}
  \node (0) at (0,0) {$1$};
  \node (1) at (0,1) {$2$};
  \draw[-] (0) edge (1);
 \end{tikzpicture}
\]
\end{document}
---------------------------------------------------------------------------------------------------------------------------------

Attachment: test.lyx
Description: application/lyx

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