how to use a relative path with form API image_button?
You need to define an absolute path, a relative path won't work. But you can use the drupal_get_path() method to retrieve the path to your module, so you don't have to worry about where it will end up in the filesystem.

For example:

$image_path = drupal_get_path('module', 'mymodule') . '/images/show-my-score-button.png';