It's not the length of the layover that matters. Whether you just change planes in the intermediate city or stick around a while shouldn't make much difference, or possibly it would cost you more to stick around a while.
What you're hoping to find is an unpopular city B (relative to the flight capacity in and out) so that flights A-B, B-C are cheaper than flights A-C. This is generally the case when A and C are wildly popular places like London and New York. It's less likely to be the case when A or C is already kind of a backwater. Of course, if A and C are TOO popular then everyone has capacity between them and there may be no room to save money.
When I was looking for flights from Toronto to Amsterdam (YYZ-AMS) I recall it was about a 30% saving to go through Iceland. In the end I valued my time higher and took a direct flight. For a recent conference, routing Toronto-Sofia (YYZ-SOF) through Istanbul instead of the usual European hubs (Paris CDG, Frankfurt, etc) cut the price in half.
I think in your case there is no substitute for searching. Any "find a flight" website will offer you both direct and changing-planes versions of your flights: Expedia, Kayak, whatever the cool kids use these days. If it starts offering you one-stop flights that are the same price or slightly cheaper than the direct, go to that airline's web site and try pricing a multi-city with a delay in B, or call that airline and ask if that would affect the price.