How do I make partially shaded nodes with tikz without hardcoding coordinates?

You can load backgrounds library and use the node anchors like a.225. The radius should be half of minimum size. Also, use \fill instead of \filldraw.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{backgrounds}
\begin{document}
\begin{tikzpicture}[mynode/.style={circle,draw=black,minimum size=10mm}]
    \node (a) at (0,0) [mynode] {\Large $x$};
    \begin{scope}[on background layer]
      \fill[fill=gray!25] (a.225) arc [start angle=225, end angle=405, radius=5mm];
    \end{scope}
\end{tikzpicture}
\end{document}