At first you should have an action (Managed Bean) object with one of these scopes :
1- conversation
2- session
3- application

and then you could use ajax base thechnology for better performance and efficiency for Rerendering page  without refresh it like Richfaces.

So then you should bind a field or variable from that action (Managed Bean) with EL to JSF page at combo box that you use like this :

<h:selectOneMenu id="dropdown-1" value="#{action.field}">
<f:selectItems value="#{helper.items}"/>
</h:selectOneMenu>

and in this case value of selected  will survive till action is live at those scope.