For buttons created with .Button() in jQuery........
Whilst the other answers will change the text they will mess up the styling of the button, it turns out that when a jQuery button is rendered the text of the button is nested within a span e.g.
<button id="thebutton">
<span class="ui-button-text">My Text</span>
</button>
If you remove the span and replace it with text (as in the other examples) - you'll loose the span and associated formatting.
So you actually need to change the text within the SPAN tag and NOT the BUTTON!
$("#thebutton span").text("My NEW Text");
or (if like me it's being done on a click event)
$("span", this).text("My NEW Text");