Depends a bit on the fare rules, but in essence a connection / layover of under 24 hours probably won't result in higher charges, though BA.com tends to not like connections that long, and occasionally there will be additional taxes to pay. If you go for a stopover of more than 24 hours it depends on the fare details, sometimes it is free of charge (though again there may be a difference in taxes, fees and charges), sometimes there is a fixed rate for each stopover, but so long as you pay that fee you can stay up to theoretically 365 days, but usually it has to be less in order that you return home within 365 days of the first flight.

If you are staying more than 24 hours you may need either assistance from someone knowledgeable, or to talk to a skilled agent / travel agent. Sometimes, for example, you may find it better to buy a ticket which isn't the cheapest but allows more flexibility (there again, some of BA's cheapest fares occasionally have generous stopover rules). On other occasions you may need to break the ticketing with open jaw sectors and separate standalone tickets to carry the gap.

And to reiterate, BA.com isn't great with complexity in this space, so if you can't get it to do what you want, you probably need to call BA.