Thanks to SharePoint Hillbilly for posting an accordion solution for SharePoint web parts for 2013.  This one right here will work on SharePoint 2010.  Read this post first to familiarize yourself.

	<!-- Reference the jQueryUI theme's stylesheet on the Google CDN. Here we're using the "Start" theme -->
	<link type="text/css" rel="stylesheet" href="//ajax.googleapis.com/ajax/libs/jqueryui/1.10.0/themes/start/jquery-ui.css" />
	<!-- Reference jQuery on the Google CDN -->
	<script type="text/javascript" src="//ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
	<!-- Reference jQueryUI on the Google CDN -->
	<script type="text/javascript" src="//ajax.googleapis.com/ajax/libs/jqueryui/1.10.0/jquery-ui.min.js"></script>
	<script type="text/javascript">
	jQuery(document).ready(function($) {
	//Put the WebPart Title for all the Web Parts you wish
	//to put into the jQuery UI Accordion into the array below.
	Accordion(["Web Part Title 1","Web Part Title 2","Web Part Title 3"]);
	});
	function Accordion(webPartTitles)
	{
	for(index in webPartTitles)
	{
	var title = webPartTitles[index];
	$("#accordion").append('<h3>'+title+'</h3>');
	$("#accordion").append('<div id="Tab'+index+'"></div>');
	$("span:contains('"+title+"')").each(function(){
	if ($(this).text() == title){
	var webPart = $(this).hide().closest("span").closest("[id^='MSOZoneCell_WebPart']");
	if ($(webPart).contents().html() != undefined)
	{
	webPart = $(webPart).contents().appendTo($("#Tab" + index));
	}
	}
	});
	}
	$("#accordion").accordion({ heightStyle: "content" });
	}
	</script>
	<div id="accordion"></div>

view raw jaccordion.html hosted with ❤ by GitHub