Adding it to the search dropdown shouldn't be difficult at all, simply open the header.tpl file and then locate the search dropdown code:
{if $search_link}
<div id="search_popup" style="display: none">
<form method="post" action="{ubb url=""}">
<input type="hidden" name="ubb" value="dosearch" />
<input type="hidden" name="daterange" value="0" />
<input type="hidden" name="where" value="bodysub" />
<input type="hidden" name="fromsearch" value="1" />
<input type="hidden" name="checkwords" value="1" />
<table class="popup_menu">
<tr>
<td nowrap='nowrap' class="popup_menu_header">{$lang.SEARCH_HEADER}</td>
</tr>
<tr>
<td class="popup_menu_content noclose">
<label>
<input type="text" name="Words" class="form-input" />
<input type="submit" name="textsearch" value="{$lang.GO_BUTTON}" class="form-button" />
</label>
</td>
</tr>
<tr>
<td class="popup_menu_content"><a href="{ubb url="ubb=search"}">{$lang.ADVANCED_BUTTON}</a></td>
</tr>
</table>
</form>
</div>
<script type="text/javascript">
registerPopup("search_popup");
</script>
{/if}
Then, add a new line after:
<tr>
<td class="popup_menu_content"><a href="{ubb url="ubb=search"}">{$lang.ADVANCED_BUTTON}</a></td>
</tr>
Like:
<tr>
<td class="popup_menu_content"><a href="{ubb url="ubb=google"}">Google Search</a></td>
</tr>