You can try the edit_comment hook that’s fired within the wp_update_comment() function, called by the edit_comment() function when you edit a comment in the backend.
This hook is fired after the comment is updated in the database.
Example:
add_action( 'edit_comment',
function( $comment_ID )
{
// ... your code here ...
}
);
Notice the following comment in the source:
The hook also fires immediately before comment status transition hooks
are fired.
I find this comment somewhat confusing, because it’s actually the get_comment filter that’s fired immediately before the comment status transition hooks are fired within the wp_transition_comment_status() function.
But it’s true if hook = action, but I usually understand the meaning of the word hooks to be both actions and filters. But this was just a minor detour.
ps: My first thought/guess was save_comment, before I checked the source 😉