You are meant to book the flights with the stopover duration you want, then contact after booking the flights to arrange the other aspects of the stopover - hotel, car hire, excursions.
Please remember to include the total duration of your stopover in your itinerary when you purchase your flight to avoid unnecessary fees or charges.
http://www.emirates.com/english/dest...w_to_plan.aspx
That's what they actually mean by "contacting your travel agent or Emirates office", I later found. But they say that in a sentence that says "booking your Dubai stopover is simple, just...". To me, booking a stopover means booking a stopover, not making hotel or car reservations or whatever.