Custom field for block's edit form? I would like to add an additional field to block's edit screen and be able to retrieve it's value upon rendering. Is it possible?
The Drupal Examples modules has an example of adding extra configuration to blocks. However, you will responsible for handling the data and how it's saved to the database which means it can get complicated fast. If it is a field with a value that is used on each of the blocks then the example which uses variable_set() and variable_get() will work. You do not want to much data saved to the variable table.

If the data needs to be more dynamic and you are not building a site with 2,000,000 visitors a day, you can make a node type and use Views to render the values in a block.