If I understood your question correctly, you have two options:
A) Use a _preprocess_block()
function:
function THEME_preprocess_block(&$vars) {
// Make sure that this is a custom block, and not a block that is provided by
// a module.
if ($vars['block']->module == 'block') {
// Load the block information.
$info = block_custom_block_get($vars['block']->delta);
// Add block description to the list of classes.
$vars['classes_array'][] = $info['info'];
}
}
B) Use the Skinr module. Probably will require more work, but it's much more flexible and editors / non-tech staff will be able to define styles of blocks. From the project page:
Skinr's main purpose is to allow the theme to define a set of reusable and modular CSS styles, and to make those styles available in Drupal's UI.