You should wrap your text in <label> elements:
<div id="TheOptions">
<input type="radio" id="test1" name="mylist"/><label for="test1">option 1</label>
<input type="radio" id="test2" name="mylist"/><label for="test1">option 2</label>
<input type="radio" id="test3" name="mylist"/><label for="test1">option 3</label>
<input type="radio" id="test4" name="mylist"/><label for="test1">option 4</label>
</div>
<input type="button" id="btnChnage"/>
Now your text is explicitly associated with the corresponding radio button and users will even be able to click on the text, visually impaired users will also be quite thankful for the improved usability. Then you can just do:
$(document).ready(function(){
$('#btnChange').click(function(){
$('label[for=test1]').html('best option');
});
});