[inebriate]

in skin1/customer/main/welcome.tpl,
before the featured_products.tpl include,
add

{include file="customer/main/categories_welcome.tpl"}
<br />

create a new file called categories_welcome.tpl in skin1/customer/main
with (change number of columns accordingly)

{capture name=dialog}
<table width="100%" cellpadding="0" cellspacing="0">
<tr>
	<td>
<table width="100%" cellpadding="5" cellspacing="1">
<tr>
{assign var="width" value=20}
{foreach from=$categories item=ca name=cats}
{if $smarty.foreach.cats.index is div by 4}
	</tr>
	<tr>
{/if}
<td width="{$width}%" class="PListCell">
<div>
<a href="home.php?cat={$ca.categoryid}" class="VertMenuItems"><img src="{if $current_category.icon_url}{$current_category.icon_url}{else}{$xcart_web_dir}/image.php?id={$ca.categoryid}&amp;type=C{/if}" /></a>
</div>
<div>
<font class="CategoriesList"><a href="home.php?cat={$ca.categoryid}" class="VertMenuItems">{$ca.category}</a></font><br />
</div>
</td>
{/foreach}
</table>
	</td>
</tr>
</table>
{/capture}
{include file="dialog.tpl" title=$lng.lbl_categories content=$smarty.capture.dialog extra='width="100%"'}

it may contain a few bugs (i found one where if you have category images displayed on the side menu they will all display the same icon once you click on a category), but it should get you started


a sample image of it can be found in the attached thumbnail