How to place a button in the center of div of dynamic size

This achieves what you're looking for in a simple, succinct, way:

.button {
    display:block;
    margin: 0 auto;
}

You need to give the button a specific width and then you can use automatic margins

.button {
  display: block;
  margin: 0 auto;
}

http://fiddle.jshell.net/CrHyd/