Blocks are considered configuration, not a content. Thus, there is no separate privilege you can give your "normal users" to allow them to edit blocks.

There is another way round. You can make some of content to be a blocks with [Nodeblock][1] or one of [it's competitors][2]:

> This module allows you to specify content type(s) as being a block. This allows the content managers of the site to edit the block text and title without having to access the block administration page. Users only need edit access to that node in order to edit it.

That way you can create content type Block, and apply node permissions to it. All of them, including "edit own block" or "edit any block".

Just be sure no one have "delete" permissions on them, as it would remove block you placed from your page, and that should be admin-only task.


  [1]: https://drupal.org/project/nodeblock
  [2]: https://groups.drupal.org/node/93499