A few airlines actually promote no-cost, or minimal-cost, stopovers at one or more of their hub airports:
-- Back when the 707s couldn't make it nonstop from California to Australia, Qantas offered no-charge stopovers in Honolulu. I did that on my first trip to Australia in 1958 -- it was also my first visit to Hawaii. Now, although Qantas flies nonstop, Hawaiian features Honolulu stopovers for travelers who use its flights from the mainland to Hawaii to connect with its new flights from Honolulu to Australia and Asia.
-- Singapore has promoted cut-priced stopovers in Singapore for many years, available to travelers on some economy fares. Similarly, Emirates promotes Dubai stopover packages. Turkish Airlines offers free local sightseeing tours on an Istanbul stopover.
But you don't have to find an airline that actively promotes stopovers. Some European airlines allow home-city stopovers on long-haul itineraries that connect through their hubs. These are often limited to certain routes, but they're available. Also, you can often find cases where you buy two separate segments for only a few dollars more than the through fare. The best approach is to start by first finding the fare on a through itinerary with a connection but no stopover. Then enter a multistep itinerary that includes the stopover you want, and see how much more you might have to pay.
Round-the-world specialist BootsnAll (bootsnall.com) recently issued a free downloadable report on "The Stopover Secret," showing the results of tests that found some cases where adding the stopover actually reduced the total price. (Yes, airfares are weird!) BootsnAll recommends -- and I heartily agree -- that you check both an airline with its base in the city where you want to stop over and one of the better independent airfare search systems, such as Kayak or Vayama. Every airline and third-party site I've checked provides for multistep trips: Just look for the option on the home page.
As still another approach, international ticketing rules for connecting flights generally require that your ongoing connection be booked on a flight leaving within four hours of your arrival at the connecting point or on the next available flight, whichever is later. On a route where the ongoing connection may not operate every day, you have the option of booking the first leg on a flight that arrives a day or two before the next departing flight, giving you a de facto stopover at no cost. Of course, this works only for routes where you can find an infrequent ongoing schedule.
The most difficult stopovers I know are opposite-coast stops in the United States. On a flight from the East or Midwest, for example, you'll have a tough time finding a cheap West Coast stopover; travelers from the West to Europe face the same problem with a possible New York stopover.
If you have the travel time, a stopover is a great way to ease the jet lag and other problems of a really long flight itinerary. Give it a try sometime.
(Send e-mail to Ed Perkins at eperkins(at)mind.net. Perkins' new book for small business and independent professionals, "Business Travel When It's Your Money," is now available through http://www.mybusinesstravel.com or http://www.amazon.com)