A simple
$('.ui-autocomplete-input').css('width','300px')
works (I tried it on the linked page with firebug) to change the first one on the page.
You can do something like:
$($('.ui-autocomplete-input')[N]).css('width','300px') #N is the nth box on the page
To change the Nth one.
To find a specific one by a characteristic, you could do many ways.
Find it by the first "option" (in this example "asp"):
$('.ui-autocomplete-input').map(function(){if ($(this).parent().children()[1].options[0].text == 'asp'){ $(this).css('width','300px'); return false;} })
Find it by it's "label":
$('.ui-autocomplete-input').map(function(){if ($($(this).parent().children()[0]).text() == "Your preferred programming language: "){ $(this).css('width','300px'); return false;}})
etc...
I'll update if you have an idea of how you want to find your combobox.
EDIT FOR COMMENT
oo, that makes it even easier. From the example source you linked to the html is already wrapped in a div.
<div class="ui-widget" id="uniqueID">
<label>Your preferred programming language: </label>
<select>
<option value="a">asp</option>
<option value="c">c</option>
<option value="cpp">c++</option>
<option value="cf">coldfusion</option>
<option value="g">groovy</option>
<option value="h">haskell</option>
<option value="j">java</option>
<option value="js">javascript</option>
<option value="p1">perl</option>
<option value="p2">php</option>
<option value="p3">python</option>
<option value="r">ruby</option>
<option value="s">scala</option>
</select>
</div>
I would give that div a unique id then:
$('#uniqueID > input.ui-autocomplete-input').css('width', '300px')
That selects child elements of your div that are inputs with a class of "ui-autocomplete-input".