[vasilis]

The template file responsible for the search box is customer/search.tpl template file:

<div class="search">
  <div class="valign-middle">
    <form method="post" action="search.php" name="productsearchform">

      <input type="hidden" name="simple_search" value="Y" />
      <input type="hidden" name="mode" value="search" />
      <input type="hidden" name="posted_data[by_title]" value="Y" />
      <input type="hidden" name="posted_data[by_descr]" value="Y" />
      <input type="hidden" name="posted_data[by_sku]" value="Y" />
      <input type="hidden" name="posted_data[search_in_subcategories]" value="Y" />
      <input type="hidden" name="posted_data[including]" value="all" />

      {strip}

        <input type="text" name="posted_data[substring]" class="text{if not $search_prefilled.substring} default-value{/if}" value="{$search_prefilled.substring|default:$lng.lbl_search|escape}" />
        {include file="customer/buttons/button.tpl" type="input" style="image"}
        <a href="search.php" class="search">{$lng.lbl_advanced_search}</a>
      {/strip}

    </form>

  </div>
</div>

In red color is the code for the display of the input text control. I guess you could replace the text type input with a select input (or add beside it so you could have both a text input and a drop-down input), something like:

<select name="posted_data[substring]" class="text{if not $search_prefilled.substring} default-value{/if}" value="{$search_prefilled.substring|default:$lng.lbl_search|escape}" />
          <option>option 1</option>
          <option>option 2</option>
          <option>option 3</option>
           ..................................
           ..................................
 </select>

I hope, it works out for you...