Dropbox is very well aware of this idea. They have made the explicit decision not to go this route. If you want this feature you can implement it for yourself without much trouble:

http://www.dropboxwiki.com/Sync_Other_Folders

But you're not who dropbox cares about.

Have you tried BitTorrent Sync? It doesn't have an always available web interface but does syncing between two (or more) online machine's directories quite well.. as long as they both have connectivity at the same time.

Try bitsync for arbitrary directory syncing.

SugarSync - https://www.SugarSync.com allows you to sync any folder you want