<button href="#" data-dropdown="drop1" aria-controls="drop1" aria-expanded="false" class="button dropdown">Order list by: </button><br>
<ul id="drop1" data-dropdown-content class="f-dropdown" aria-hidden="true">
    <li><a href="#">This is a link</a></li>
    <li><a href="#">This is another</a></li>
    <li><a href="#">Yet another</a></li>
</ul>

$('li').on('click', function(){
    $('button span').text($(this).text());
});

<button href="#" data-dropdown="drop1" aria-controls="drop1" aria-expanded="false" class="button dropdown">Order list by: <span id="orderBy"></div></button><br>

$("li a").on("click", function() {
    $("button #orderBy").text($(this).text());
});