Inexpensive air fare - travel roulette

You can do this up to seven days in Reykjavik with Iceland Air

I don't know of a website that highlights layover possibilities in an easy-to-research way, but there may be one. Certainly Turkish Air has a lot of trips that give you a chance for a quick stop in Istanbul. Frankly, those things don't appeal to me, because I'd want more than one brain-dead day at my landing point, so I read right past those itineraries when they pop up in my flight searches.

If your stopover city is one with comparatively inexpensive flights from your origin (that often includes Reykjavik, London, Dublin), you could expand your consideration by planning at least one additional day there (for protection against inbound flight delays) and adding on a budget one-way flight to a continental destination of your choice. Much wider variety of ultimate destinations that way, I would think.

This would mean a multi-city flight to Europe (into Reykjavik, London, or Dublin; back from wherever), or else you'd need to do another budget hop at the end of your trip, back to your origin. In the latter case, you'd again need time there, because your return flight home would be hanging in the balance. Missing it wouldn't be a pleasant experience, economically.

But don't forget to consider the cost of checking a bag on the budget-airline flights if that's something you need to do, and pay attention to the airports being used. In some cases they are obscure ones far from the city and without convenient, cheap public transportation.

One possible approach, admittedly not very efficient, might be this: Choose the furthest away city, hopefully a big one, and use an agency or matrix.itasoftware.com to see all the routes from your home city. This will show what airlines require transfers and in what cities.
Then start over, using a multi-destination search fuction, to enter your departure, the middle city you want to see, and the destination. This will, at least, probably be cheaper than buying each leg separately.

Here's a good link with information on stopovers. There are some additional airlines like EVA (Taipei), Korean, ANA, and Aer Lingus that also offer some limited stopover programs. They recommend Skyskanner and Airwander for scoping out different destinations and airfare. I have also found Google Flights good for this. If you put in your originating city and leave the destination blank, you can click on the map just below the flight information and explore different destinations. On the map, you can click on Nonstop on the upper left menu will show you all nonstop flights from your originating city.

I have been flying into cities that are not my final destination for several years. Most recently, Dublin then on to Madrid via Ryanair. It is very feasible to do this because most anywhere in Europe has low cost airlines to get to another destination. Next trip to Italy I will fly to Zurich spend the night and fly into Rome. This is a wonderful way to get a taste of a city. But I also use this a strategy to save my miles. Dublin and Zurich are half the miles for a business class seats than flying directly to Madrid or Rome. So essentially gives me more business class trips with my miles.

Yes, we've done this both times we've gone to Sicily. I find the city that's the cheapest to fly into from my home base (one year it was Brussels, one year Frankfurt), go there for a night or two, and then take an intra-European flight to our final destination.

I do that all the time it helps to be flexible and not scared about stuff.
For example I only had enough miles last summer to go one way Chicago to Zurich.
I ended buying one way ticket back on Wow Air from Paris for about $ 360
There is no site special site but use google flights

OOOH, just went to Goggle Flights - thats great .. so fun. Thanks (I m thinking it might be Stockholm for a couple of days, but Amsterdam is in running)
thanks

Skyscanner also lets you select 'Everywhere' (or maybe it's 'anywhere') as a destination once you enter a starting point.