searchform.html

<div id="searchbox">
  <h2 class="search"><%= label_tag(:q, "Search ") %></h2>
  <% form_tag root_path, :method => :get do %>
    <span id="search_box_and_link">
      <%= text_field_tag(:q, localized_params[:q], :class=>"search") %>
    </span>
    <noscript>
      <%= label_tag(:search_field, " in ") %>
      <%= select_tag(:search_field, options_for_select(search_fields, h(localized_params[:search_field])), :title => "Targeted search options") %>
    </noscript>
    <span id="split_button" style="display:none;">
      <div class="button_set">
        <button id="run">Search</button>
    	<button id="select">select</button>
      </div>
    </span>
    <%= hidden_field_tag(:hidden_search_field, h(localized_params[:search_field])) %>
    <noscript>
      <%= submit_tag 'search', :class=>'submit' %>
    </noscript>
  <% end %>
</div>
<script type="text/javascript">
  // your jQueryUI button() and buttonset() functions
</script>

your.js

$(document).ready(function() {
  // Display the split button
  $("#split_button").toggle();
  // change the name of the hidden_search_field so the form sends the param along properly for JS enabled browsers, but defaults to the drop down for non-js browsers.
  $("#hidden_search_field").attr("name","search_field");
});

  // in the onclick event for the item in your drop down.  You will need some way of determining what the value to pass for the search_field param is.  I put the value that I want to pass as the search_field parameter as the class of the element that the user is clicking on.
  $('#hidden_search_field').attr('value',$(this).attr("class"));