This is pretty easy...

Assuming you already have a rich:suggestionbox component:

<rich:suggestionbox id="mySuggestionBox"
   for="myInputBox"
   suggestionAction="#{MySuggestionBean.getSuggestions}"
   var="suggestionWrapper"
   fetchValue="#{suggestionWrapper.plainSuggestion}"
   onselect="a4jSuggestionSelected();"
   minChars="1">
   <t:column><h:outputText value="#{suggestionWrapper.formattedSuggestion}" escape="false"/></t:column>
</rich:suggestionbox>

You just create an image to the right of your input box:

<h:graphicImage url="/images/suggestion-box-down-arrow.png"
   onclick="toggleSuggestions(#{rich:component('mySuggestionBox')});"
   alt="Show list"
   styleClass="suggestionBoxDownButton"/>

And your JavaScript method:

function toggleSuggestions ( suggestionBox ) {
   if( suggestionBox.active )
       suggestionBox.hide();
   else
       suggestionBox.callSuggestion( true );
}

Voila!

Hope this helps somebody.

-Shadow

richfaces-showcase online sample(second sample on the page)