Each category item has an ID number. You can add background images to the categories in the widget and then do some trickery set a negative left margin (-999px) to move the text far off to the left and out of site.
You would create new rules for each category in the CSS using the classes that are set for each category. Example: cat-item-10151 is your At Home category. You would create rules for each individual category. You would upload your images to the media library in the size needed, which appears to be around 35px in height. This could be adjusted to accommodate larger images with a little more CSS work.
I've not tried any of the below, but I am pretty sure it will work for you..
.cat-item-10151 {
background: url("URL OF THE ICON") no-repeat scroll 0 0 transparent;
}
.cat-item-10151 a {
margin-left: -999px;
}
Under settings > sharing, you can add your own sharing services and use new images. Basically you would be creating new services to replace the existing.
http://en.support.wordpress.com/sharing/#adding-your-own-service