"just right" integral sign for tikz integrator block

A \scalebox should suffice.

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{shadows, positioning}
\begin{document}
\begin{tikzpicture}[node distance=5mm, auto, font=\sffamily,
       blockcolors/.style={
        % The rest
        thick,draw=black,
        top color=white,
        bottom color=black!10,
        font=\sffamily\small
    },
    blockheight/.style = {
        minimum height=10mm
    },
    block/.style={
        % The shape:
        rectangle, minimum size=6mm, minimum width=12mm,
        blockheight,
        node distance=5mm,
        blockcolors,
        drop shadow
    }
    ]
\node (int1) [block, label=below:too small]{$\int dt$};
\node (int2) [block, right=of int1, label=below:too big]{$\displaystyle\int dt$};
\node (int3) [block, right=of int2, label=below:just right]{$\scalebox{1.4}{$\int$}dt$};
\end{tikzpicture}
\end{document}

With \usepackage{nccmath} you can type

$\medint\int dt$

in the third box and the result will be

Goldilocks is wrong here: the right one is in the middle.