I know your question asks re a 12cm fan, but that big may not be necessary. See how hot it gets.
If you did want 12cm, you could power it from the mains/acdc.
For example any USB-SATA/USB-IDE adaptor comes with a plug with something like a 4 pin molex or similar,like a sata power connector. So, either way, with 12V, and the fan can connect to that whole plug, it may need an adaptor but they're easy to get.
Laptops tend to have small cpu fans. alternatively, there is this option(I see it titled mini vacuum air extracting usb case cooler cooling fan). Looks like it goes adjacent to the vent. But since you have just a motherboard you could blow it elsewhere. Just onto the hot part.
If there's a metal plate over the CPU that takes heat, then you would blow onto that.
I'm not sure how you'd do that with a blower on the side - maybe build a tube to redirect the air (let me know what you'd use to do that if you do). Assuming the CPU is on the bottom of the motherboard, you could raise the laptop a bit and blow the air under it from the side that may do it. Or, use a fan below it as you seem to plan to do.
You can also get a more typical laptop cooler, a laptop cooling pad, some have a big fan like 14cm.. pluscom do a 14cm one and they do a 12cm one.. The description says Aluminum USB Notebook / Laptop Cooler 12cm Fan with speed control Switch & 2 USB ports So, it is able to get its power from 2 USB ports. You can get ones with 3 small fans too. All USB powered no ACDC adaptor.