A checkbox in the options menu, stored in a cookie. Wherever its generating the thread or whatever check for the flag and either use western or japanese order for the name.
Actually maybe not, because it probably picks the name whenever you insert a new row into whatever Post or Thread database table it uses.
I guess it could still work but you might accidentally swap the name order of someone who manually uses a characters name in Western order, but that's probably not too big of a deal.
$japOrderCookie && $NAMES->get(post->name) ? echo $JAPORDER_NAMES->get(post->name) : echo $WESTORDER_NAMES->get(post->name) ?>
I'm not actually sure if thats valid php but something like that could work.