Assuming the motherboard is proven to be the faulty component, there are a couple of easily salvaged parts. These are the CPU and RAM. If you want to expand beyond the motherboard, you could also imply that the hard drive, power supply, and case fans are all still good. Any cards plugged into PCI slots could also be pulled and used. If you really need to dig deep, you could pull out screws, jumper pins, and even the CMOS battery.
The only precautions are to ensure CPUs and RAM used on different motherboards are compatible with socket type, and for RAM, that the new motherboard is able to read the speed (for example, trying to put 1600 RAM in a board that only supports up to 1333 may cause problems).
If you mean more advanced parts, such as transistors, heat sinks, or other small parts directly soldered to the board, your guess is as good as mine. In the IT world, where I once worked professionally, we would rather splurge $50 on a new motherboard than spend time trying to pull much from a computer. The cost usually outweighs the benefit of trial and error with testing and using old parts.