Center a Node Vertically TikZ

You could use the positioning library and then use the left=of <node identifier> for positioning the node containing the array; I added some more files to the array to see the effect:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning}

\begin{document}

\begin{tikzpicture}
\setlength{\arraycolsep}{2pt}
\node[draw,fill,circle,inner sep=1pt,label=right:$C$] at (3,0) (c) {};
\node[align=left,rounded corners,fill=black!10,inner sep=1ex,left=of c]
  {$
    \left\{
      \begin{array}{llc}
        h &=& \overline{AF}\\
        l &=& \overline{BF}\\
        l'&=& \overline{AB}\\
        d &=& \overline{CE}\\
        h &=& \overline{AF}\\
        l &=& \overline{BF}\\
        h &=& \overline{AF}\\
        l &=& \overline{BF}\\
      \end{array}
    \right.
  $};
\end{tikzpicture}

\end{document}

More generally (as Caramdir has suggested), you can use left=of current bounding box.center:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning}

\begin{document}

\begin{tikzpicture}
\setlength{\arraycolsep}{2pt}
\node[draw,fill,circle,inner sep=1pt,label=right:$C$] at (3,0) (c) {};
\node[align=left,rounded corners,fill=black!10,inner sep=1ex,left=of current bounding box.center]
  {$
    \left\{
      \begin{array}{llc}
        h &=& \overline{AF}\\
        l &=& \overline{BF}\\
        l'&=& \overline{AB}\\
        d &=& \overline{CE}\\
        h &=& \overline{AF}\\
        l &=& \overline{BF}\\
        h &=& \overline{AF}\\
        l &=& \overline{BF}\\
      \end{array}
    \right.
  $};
\end{tikzpicture}

\end{document}

It's possible to avoid the positioning library :

1) Align with (C)

 \node[align=left,rounded corners,
       fill=black!10,inner sep=1ex,anchor=east] at ([xshift=-2cm]c) {....

2) Align with (current bounding box)

  \node[align=left,rounded corners,fill=black!10,
        inner sep=1ex,anchor=east] at (current bounding box.west) {...