Adjusting font size with TikZ picture

You can change the font size inside a tikZ node like you do it in normal LaTeX – use one of these:

\tiny
\scriptsize
\footnotesize
\small
\normalsize
\large
\Large
\LARGE
\huge
\Huge

e.g.

\node (c) at (1,2) {\large x};

or

\node (c) at (1,2) {\large $x$}; %note the \large *outside* the inline math

Edit: To change font size inside math mode, LaTeX provides the following commands:

\displaystyle
\textstyle
\scriptstyle
\scriptscriptstyle

Using scaling algorithms provided by tikZ/pgf (e.g. scale=...) scales "the entire character", so it may look ugly if you use too much scaling. If you set the font sizes with the above commands, LaTeX selects different symbols for the different font sizes. This ensures the fonts are readable and have enough "details". If you want a more extreme scaling, use the scale=3.0 option for the node.

Set a new style so that you can use it later.

For example:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{relsize}

\tikzset{fontscale/.style = {font=\relsize{#1}}
    }

\begin{document}
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,3);
  \coordinate (a) at (rnd,rnd);
  \coordinate (b) at (3-rnd,3-rnd);
  \draw (a) -- (b);
  \node (c) at (1,2) [fontscale=4] {x};
  \draw let \p1 = ($ (a)!(c)!(b) - (c) $),
            \n1 = {veclen(\x1,\y1)}
        in circle [at=(c), radius=\n1];
\end{tikzpicture}
\end{document}

At the top, write

\begin{tikzpicture}[thick,scale=1, every node/.style={scale=1.3}]
\draw [help lines] (0,0) grid (3,3);
\coordinate (a) at (rnd,rnd);
\coordinate (b) at (3-rnd,3-rnd);
\draw (a) -- (b);
\node (c) at (1,2) {x};
\draw let \p1 = ($ (a)!(c)!(b) - (c) $),
        \n1 = {veclen(\x1,\y1)}
    in circle [at=(c), radius=\n1];
\end{tikzpicture}

the thick changes your arrows, the first scale changes the scale of your drawing, but the second argument changes the size of your nodes, presumably where you have your text. This will change for all nodes.