Is it possible to filter the display name for post formats for display in the Formats meta box?

Yes, you can. You can hook right into gettext. The example provided below changes Publish button:

add_filter( 'gettext', 'binda_change_publish_button', 10, 2 );

function binda_change_publish_button( $translation, $text ) {
    //check if this is pizza add or edit page in administration 
    global $pagenow, $typenow;
    if ( is_admin() && ( $pagenow == 'post-new.php' or $pagenow == 'post.php' ) && ( $typenow == 'your_post_type' or ( isset($_GET['post_type']) && $_GET['post_type'] == 'your_post_type') ) ) {
        if ( $text == 'Publish' )
            return 'Save My post type';     
    }
    return $translation;
}

EDIT: Tried to change Aside to “Postranní” (in Czech) and it works without problem when using another filter gettext_with_context. See example below:

add_filter( 'gettext_with_context', 'binda_change_standart_format_name', 10, 2 );

function binda_change_standard_fromat_name( $translation, $text ) {
    //check if this is pizza add or edit page in administration 
    global $pagenow, $typenow;
    if ( is_admin() && ( $pagenow == 'post-new.php' or $pagenow == 'post.php' ) && ( $typenow == 'post' or ( isset($_GET['post_type']) && $_GET['post_type'] == 'post') ) ) {
        if ( $text == 'Aside' )
            return 'Postranní';     
    }
    return $translation;
}