I am \"creating\" my own \"ComboBox\" using Bootstrap 3 and JavaScript. Here is the JSFiddle for what I have so far:
You might want to modify your jQuery code a bit to '#demolist li a' so it specifically selects the text that is in the link rather than the text that is in the li element. That would allow you to have a sub-menu without causing issues. Also since your are specifically selecting the a tag you can access it with $(this).text();.
$('#datebox li a').on('click', function(){
//$('#datebox').val($(this).text());
alert($(this).text());
});