The cost is, at a minimum, the cost of the socket connector, the compression mounting and any related standoffs, screws and mountings to hold the heatsink in place.
An 1156 compatible CPU socket is PE115627-4051-01F which, depending on quantity can be bought on Alibaba for anywhere between $0.10 to $10.
Then you need some compression mounting arrangement to hold the CPU in place, probably a lever fit, I'd probably say it's in the same ball park as the CPU socket.
For a laptop the heatink and fittings will likely be the same (or very similar) either way.
Going for a BGA over a socketed processor will massively simplify the mechanical design of a laptop, saving (potentially) a few hundred man-hours of labour. It will also simplify the build of the laptop as the CPU can be fitted at the same time as all the surface mount components, instead of needing a second operation after the socket itself is soldered.
For the manufacturer all this builds up.
So, maybe $20 per PCB for a CPU socket and mechanical fittings, then maybe another $5 in time to fit and sort out the fitting of the CPU, over a cost of $0 to do it using BGA. The restraints holding the heatsink in place might be very similar between the two methods.
For a run of (maybe) 10,000 laptop motherboards I suspect that going with BGA over sockets could amount to a design time saving in the ball part of anywhere from $20,000 to $50,000, with a similar but probably more moderate saving in parts and labour time.
This is why most laptops will go for soldered components. You can easily make different variants where the only difference is the PCB with very little cost between motherboad PCB runs. Granted you make it easier to configure the processor later in the process if you go for socketed over BGA, but it costs significantly more in parts and labour to do it that way.
On a board that makes up potentially $100 of a $300 laptop, saving $20 can be very significant.