cheapest = inverter
wire depends on amount of amps drawn
Factory wiring (in my opinion) is best.
My cig lighter 2.5mm wire supplies my 60 watt rig without fear of a fire.
The head light on my car has a 55 watt bulb and the same diameter wire as the cig lighter port in the trunk.
I don't understand the overkill suggestions of massive gauge wire to the battery as the only way to wire a carputer.
I do understand the need to eliminate the cig lighter plug and solder to the wires attached there.
I have a 435 Mhz p1 with 96 ram and it runs my music and gps just fine, the movies are a little laggy but not too bad. You are gonna be putting more stress on your system with all the other stuff youll be using your computer for. My point is you might be able to make it work with an old sub p4 desktop if you streamline the operating system as much as you can.
Go to Ebay and see what you can find. stuff is always cheaper over there