If you view the source for your site,

 you can see that the links inside the table are formatted as follows:

<a href="modules.php?name=Downloads">Downloads</a><br>

A new line like that is generated for every active module.
Those lines are being generated by this code in block-Modules.php:

$content .= "<a href="modules.php?name=$m_title">$m_title2</a><br>n";

Create the images you want to use to make up your menu with your favorite paint program..
I recommend gif format, 148 wide, however tall you want.
Here is my button:

Each image must be named EXACTLY like the module name.
So, in this example, I need an image called "Downloads.gif".
Remember that the filename is CASE SENSITIVE.
Put your images in HTML/themes/yourtheme/images/blocks/
(you need to create the blocks directory)

We want the code in block-Modules.php to generate lines like this:

<a href="modules.php?name=Downloads"><img src="/themes/yourtheme/images/blocks/Downloads.gif" alt="Downloads" width="148" height="24" border="0"</a><br>

So here's the code that does that:

$content .= "<a href="modules.php?name=$m_title"><img src="/themes/yourtheme/images/blocks/$m_title.gif" alt="$m_title2" width="148" border="0"</a><br>n

replace the line in /* make modules block look like I want */ that begins with $content .=with the above code and see what happens!