Hi,
My thoughts for your consideration....
Power sockets on laptops are open to physical abuse and can break free of the motherboard. This is quite a common occurrence with heavy handed or careless users. It is worth carefully inspecting the area around the power socket as the sort of heating you describe is uncommon in good solder joints working at less than 60A ! I mention this because it would be a pity to fit the new socket before ensuring that all is well below it
Now onto your fault. You stated that the meter buzzed across all socket pads... useful but please be aware that an Ohms reading can offer more information than the <20 Ohms buzzer. A physical short circuit is not likely between power plane and 0V unless the unit has been messed with at some point. A shorted component across said power and 0V rail is far more likely as has already been stated.
In such a case I would normally break out my Polar Toneohm which is basically a smart milliohm meter that leads the user to the lowest impedance across the power rails and so to the shorted component. Unfortunately these are expensive and normal multimeter's will not do the same job. I have been thinking laterally for you in terms of basic physics.
1. You could buy some freezer spray from Maplin (~GBP5) and spray it on the area of interest, say 10cm x 10cm around the power socket. Let it frost over, then apply power to the board from the laptop PSU. The short should cause a heating effect in the area causing the problem and also possibly on the feed tracks to it. The frost should rapidly disappear at those points and may assist you with tracing the fault. The problem will be if the laptop power supply goes into foldback and current limits to a level too low to cause any heating effect. You could also try putting the PCB in the freezer overnight and seeing if a frost forms when removed...that's free
2. Buy a cheap Infra Red thermometer. I found some on ebay for around GBP10. These are like tiny single element thermal 'imagers' in that they 'see' heat and so you could apply power to the PCB and use the IR thermometer to trace over the surface of the PCB looking for an area of increased heat dissipation. This would hopefully be the location of the short as the PSU overcurrent current foldback will likely prevent the motherboard powering up to any extent and so little, if any IC activity will be generating heat.
3. MK1 eyeball, bright light and a 5X or 10X loupe.... its free and can often spot areas of localised heating. With all the heat generated at the power socket I would expect some pcb lacquer discolouration at the location of the short. A time consuming search process but cheap !
I still recommend very careful examination of the area directly under the power socket. If there was a dry solder joint, it could have caused localised heating that in turn damaged the PCB and interlayer insulation...causing a short circuit at the power input point on the PCB.
Good luck.... it will be a challenge without the normal repair equipment for this type of fault.
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