Create a custom block with input format "PHP" with PHP code in it that looks something like this:
<?php $result = db_query("SELECT nid, title, uid, '' FROM node WHERE uid = ## AND type = 'blog' ORDER BY created LIMIT 10");
$content = node_title_list($result, "Joe Bob's Blog"); echo $content; ?>
You'll need to change ## to the numeric user id of the user's blog you want.