Edit in JSFiddle

/**
* Initialize customized selectbox
* @element to apply
**/
$.fn.setCustomizedSelectbox = function() {
	var $selectbox = $(this),
		$optionbox = $selectbox.children("ul.selectbox-options"),
		$options = $optionbox.children("li");
	var isOpened = false;

	function _onToggleOptionBox(event) {
		event.stopPropagation();

		var target = event.target;

		if($.inArray(target, $options) !== -1) {
			if(isOpened) return toggleOptionItem(target);
			isOpened = true;
		} else {
			if(!isOpened) return;
			isOpened = false;
		}

		$optionbox.toggleClass("opened");
	}

	function _onCloseOptionBox(event) {
		event.stopPropagation();

		var $this = $(this),
			$target = $(event.target);

		if(($.inArray(event.target, $options) !== -1 || $target.is($this)) && isOpened) {
			$optionbox.toggleClass("opened");
			isOpened = false;
		}
	}

	function toggleOptionItem(selected) {
		var $selectedItem = $(selected),
			value = $selectedItem.data();
			value = value && value.value || null;
		
		console.log(value);
		// 선택된 아이템의 값을 이곳에서 처리하면 됩니다.
		// form 에 적용한다면 hidden input box 를 만들어서 value 를 업데이트 하거나,
		// 페이지 이동이 필요하면 이곳에서 href relocation 을 처리하면 됩니다. :)

		if(!$selectedItem.hasClass("selected")) {
			$options.removeClass("selected");
			$selectedItem.addClass("selected");
			$selectedItem.trigger("onSorterSelected");
		}

		$selectbox.trigger("click");

		return;
	}

	$selectbox.on("click", _onToggleOptionBox);
	$optionbox.on("mouseleave", _onCloseOptionBox);
}

$(document).ready(function() {
	$("#selectbox").setCustomizedSelectbox();
	$("#selectbox2").setCustomizedSelectbox();
});

<div id="selectbox" class="custom-selectbox">
	<ul class="selectbox-options">
		<li class="selected" data-value="item1">아이템 하나</li>
		<li data-value="item2">아이템 둘</li>
		<li data-value="item3">아이템 셋</li>
		<li data-value="item4">아이템 오</li>
	</ul>
</div>

<div id="selectbox2" class="custom-selectbox">
	<ul class="selectbox-options">
		<li data-value="item11">선택지는 하나</li>
		<li class="selected" data-value="item12">선택지는 둘</li>
		<li data-value="item13">선택지는 셋</li>
		<li data-value="item14">선택지는 오</li>
	</ul>
</div>

.custom-selectbox {
	position: relative;
	float: left;
	width: 100px;
	height: 40px;
	margin-right: 5px;
	z-index: 10;
}

ul.selectbox-options {
	position: absolute;
	width: 100%;
	height: 40px;
	max-height: 40px;
	margin: 0;
	padding: 0;
	border: 1px solid #aaa;
	background-color: #fcfcfc;
	z-index: 9;
	transition: max-height .5s ease-out;
}
ul.selectbox-options li {
	list-style: none;
}
ul.selectbox-options.opened {
	height: auto;
	max-height: 500px;
}
ul.selectbox-options li {
	display: none;
	height: 40px;
	padding: 0 10px;
	line-height: 40px;
	font-size: 12px;
}
ul.selectbox-options li.selected {
	font-weight: bold;
}
ul.selectbox-options li.selected,
ul.selectbox-options li:first-child,
ul.selectbox-options.opened li.selected:first-child {
	background-repeat: no-repeat; background-image: url(http://rootbox.co.kr/resources/customized_checkbox/customized_selectbox.png); background-position: 100% 6px;	
}
ul.selectbox-options.opened li:first-child,
ul.selectbox-options.opened li.selected:first-child {
	background-position: 100% -22px;
}
ul.selectbox-options.opened li {
	display: block;
}
ul.selectbox-options.opened li.selected {
	background-image: none;
}
ul.selectbox-options li.selected {
	display: block;
}
ul.selectbox-options li:hover {
	background-color: #f5f5f5;
}