Getting initial selector inside jquery plugin

I got some help earlier regarding selectors, but I'm stuck with the following.

Lets say you have a plugin like this

$('#box').customplugin();

how can I get the #box as a string in the plugin? Not sure if that's the correct way of doing it, and any other solution would be great as well.

Considering #box is a select dropdown,

The problem I'm having is if I do the regular javascript

$('#box').val(x);

The correct option value gets selected,

but if i try the same inside a plugin

.....
this.each(function() {
var $this = $(this);


$this.val(x);

the last code doesn't really do anything.

I notice I'm having trouble targeting #box inside the plugin because it's a object and not a string...

Any help would be appreciated.

Thanks!

Edit:: Putting in the code I'm working in for better understanding

(function($){
$.fn.customSelect = function(options) {
    var defaults = {
        myClass : 'mySelect'
    };
    var settings = $.extend({}, defaults, options);


    this.each(function() {
        // Var          
        var $this = $(this);
        var thisOpts = $('option',$this);
        var thisSelected = $this[0].selectedIndex;
        var options_clone = '';

        $this.hide();

        options_clone += '<li rel=""><span>'+thisOpts[thisSelected].text+'</span><ul>'
        for (var index in thisOpts) {
            //Check to see if option has any text, and that the value is not undefined
            if(thisOpts[index].text && thisOpts[index].value != undefined) {
                options_clone += '<li rel="' + thisOpts[index].value + '"><span>' + thisOpts[index].text + '</span></li>'
            }
        }
        options_clone += '</ul></li>';

        var mySelect = $('<ul class="' + settings.myClass + '">').html(opti