How to mark an option as selected

What you are looking for is the selected() function.

Your updated option should look like this:

<option
    value="<?php echo $category->cat_ID; ?>"
    <?php selected($option, $category->cat_ID); ?>>
    <?php echo $category->cat_name; ?>
</option>

// EDIT
Oh, and BTW, in your particular case you might want to use the wp_dropdown_categories function, I guess.

That would be:

wp_dropdown_categories(
    array(
        'selected' => $option
        // maybe some other settings
    )
);