Update: You could put a larger bounding box around the whole diagram (I think @AndréC's option from the comments is conceptually better than the path I originally had even if the effect is the same), make the loop only go to 350° so you don't get the pause at the end, and change from a bullet in a node to directly drawing a circle. @BlackMild's suggestion to use (\angle:1)
instead of the trig functions is also good.
\documentclass[tikz]{standalone}
\begin{document}
\foreach \angle in {0,10,...,350}
{
\begin{tikzpicture}
\draw (0,0) circle (1);
\fill[blue] (\angle:1) circle (0.07);
\useasboundingbox (-1.1,-1.1) rectangle (1.1,1.1);
\end{tikzpicture}%
}
\end{document}

The animated GIF was created using ImageMagick. It's animation documentation can be found here.
I use a simple shell script for the conversation:
#!/bin/bash
BASE=`basename $1 .pdf`
PDF="$BASE.pdf"
GIF="$BASE.gif"
RESOLUTION=300
CONVERT=/usr/bin/convert
CONVERTOPTS="-density $RESOLUTION -delay 8 -loop 0 -background white -alpha remove"
echo "Converting to TSX GIF..."
$CONVERT $CONVERTOPTS $PDF $GIF > /dev/null
\useasboundingbox (-1.5,-1.5)rectangle(1.5,1.5);
:)