Edit in JSFiddle

/**
* Initialize customized checkbox
* @element to apply
**/
$.fn.setCustomizedCheckbox = function() {
	var IEVersion = getIEVersion();

	if(!IEVersion || IEVersion > 8) return; // Ignore greater than IE 8

	var $this = $(this),
		$checkboxes = $this.find(".search-customized-checkbox input[type=checkbox]");

	$checkboxes.each(function(index) {
		var $this = $(this),
			$label = $this.siblings("label").eq(0),
			$span = $("<span>").addClass("custom-checkbox");
		var isChecked = getCheckboxStatus($label);

		$span.addClass(isChecked ? "checked" : "");
		$label.addClass(IEVersion === 8 ? "hidden-for-ie8" : "");
		$label.prepend($span);
		$label.on("click", _onStatusChanged)
	});

	function getCheckboxStatus(element) {
		var $checkbox = $("#" + element.attr("for"));
		return $checkbox.is(":checked");
	}

	function getIEVersion() {
		var ua = window.navigator.userAgent;
		var msie = ua.indexOf("MSIE ");

		if (msie > 0) {
			return parseInt(ua.substring(msie + 5, ua.indexOf(".", msie)));
		} else {
			return 0;
		}
	}

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

		var $this = $(this),
			$mock = $this.children("span.custom-checkbox");
		var isChecked = !getCheckboxStatus($this);

		if(isChecked) {
			$mock.addClass("checked");
		} else {
			$mock.removeClass("checked");
		}
	}
}

$(document).ready(function() {
	$("#checkboxes, #checkboxes2").setCustomizedCheckbox();
});
<div id="checkboxes">
	<ul class="search-customized-checkbox search-option-items">
		<li class="search-option-item">
			<input type="checkbox" data-name="brands" id="optionItem1360" value="1360" checked>
			<label for="optionItem1360" class="item-name">엘룸</label>
		</li>
		<li class="search-option-item">
			<input type="checkbox" data-name="brands" id="optionItem1361" value="1360">
			<label for="optionItem1361" class="item-name">골룸골룸</label>
		</li>
		<li class="search-option-item">
			<input type="checkbox" data-name="brands" id="optionItem1362" value="1360">
			<label for="optionItem1362" class="item-name">엘리시아</label>
		</li>
		<li class="search-option-item">
			<input type="checkbox" data-name="brands" id="optionItem1363" value="1360">
			<label for="optionItem1363" class="item-name">엘룸</label>
		</li>
	</ul>
</div>

<div id="checkboxes2">
	<ul class="search-customized-checkbox search-option-items">
		<li class="search-option-item">
			<input type="checkbox" data-name="brands" id="optionItem1370" value="1360" checked>
			<label for="optionItem1370" class="item-name">엘룸</label>
		</li>
		<li class="search-option-item">
			<input type="checkbox" data-name="brands" id="optionItem1371" value="1360">
			<label for="optionItem1371" class="item-name">골룸골룸</label>
		</li>
		<li class="search-option-item">
			<input type="checkbox" data-name="brands" id="optionItem1372" value="1360">
			<label for="optionItem1372" class="item-name">엘리시아</label>
		</li>
		<li class="search-option-item">
			<input type="checkbox" data-name="brands" id="optionItem1373" value="1360">
			<label for="optionItem1373" class="item-name">엘룸</label>
		</li>
	</ul>
</div>
.search-customized-checkbox,
.search-customized-checkbox li {
	margin: 0;
	padding: 0;
	list-style: none;
}
.search-customized-checkbox input[type="checkbox"] {
	position: absolute;
	width: 1px;
	height: 1px;
	margin: -1px;
	padding: 0;
	border: 0;
	overflow: hidden;
	clip: rect(0,0,0,0);
}
.search-customized-checkbox input[type="checkbox"] + label > span.custom-checkbox,
.search-customized-checkbox input[type="checkbox"] + label:before {
	background: url(http://rootbox.co.kr/resources/customized_checkbox/customized_checkbox.png) no-repeat 0 0;
}
.search-customized-checkbox input[type="checkbox"] + label:hover > span.custom-checkbox,
.search-customized-checkbox input[type="checkbox"] + label:hover:before {
	background-position: -14px 0;
}
.search-customized-checkbox input[type="checkbox"] + label {
	position: relative;
	display: inline-block;
	padding: 2px;
	cursor: pointer;
	-webkit-user-select: none;
	-moz-user-select: none;
	-ms-user-select: none;
}
.search-customized-checkbox input[type="checkbox"] + label > span.custom-checkbox,
.search-customized-checkbox input[type="checkbox"] + label:before {
	content: '';
	display: inline-block;
	width: 14px;
	height: 14px;
	margin: -2px 8px 0 0;
	line-height: 14px;
	text-align: center; 
	vertical-align: middle;
}
.search-customized-checkbox input[type="checkbox"] + label > span.custom-checkbox.checked,
.search-customized-checkbox input[type="checkbox"]:checked + label:before {
	background-position: -28px 0;
}
.search-customized-checkbox input[type="checkbox"] + label:hover > span.custom-checkbox.checked,
.search-customized-checkbox input[type="checkbox"]:checked + label:hover:before {
	background-position: -42px 0;
}
.search-customized-checkbox input[type="checkbox"] + label.hidden-for-ie8:before {
	content: none;
	display: none;
}

@media screen and (-webkit-min-device-pixel-ratio: 1.5) {
	.search-customized-checkbox input[type="checkbox"] + label > span.custom-checkbox,
	.search-customized-checkbox input[type="checkbox"] + label:before {
		background-image:url(http://rootbox.co.kr/resources/customized_checkbox/customized_checkbox@2x.png);
		background-size: 56px 14px;
	}
}

External resources loaded into this fiddle: