How to change this "Anonymous's Blog" link to another name?
The link name is derived from the author of that node. The variable should be $content['links']['blog']['#links'] array, which can be found in node--blog.tpl.php. You can manipulate the data at that point, which is best kept as simple as possible, due to being in the tpl file itself. 

If you're looking to do complex changes, I would suggest a _preprocess_node in your template, which is probably ideal if you're doing anything more than an unset or simple truncation.

More information can be found at https://www.drupal.org/node/1783428 and Preprocess function for node.tpl.php file respectively.