Electronic computer components are sensitive to over-voltages. Voltage surges in the power supply can cause insulation break-down that keeps various parts from working. Lightning storms often cause especially-high voltage surges that can pass through many surge protectors and cause short circuits in your computer. The key is to figure out what part is damaged and replace it.
Lights and Sound
If there's no fan sound and no pilot lights come on when you switch on the computer, it may not be getting power. The problem can be with the socket, the power brick or the power cable. Plug a light into the socket to make sure there is power. If there is, plug the computer directly into the socket to bypass the power bar. If it works, replace the power bar; if not, replace the power cord. If the computer still doesn't start, the problem is inside the case.
Video of the Day
The computer power supply is the most likely component to fail during a lightning storm because the power surges hit it first. If you're comfortable working with electronics, you'll be able to change the power supply because on most models it simply mounts to the frame with four screws from the outside and is connected to the other components with plugs. Unplug everything before opening the computer case, and then unplug the connectors and note where they were connected. Unscrew the four power supply screws from outside the case and remove the power supply unit. Get an exact replacement from a local computer supply store and install it. If that was the only fault, the computer will start normally.
Video Card or RAM Failure
Sometimes the power surge damages additional components or passes through the power supply harmlessly and short circuits other electronic cards. If, with the computer open, you switch it on and see lights on the motherboard or hear drives operating, the power supply is probably not damaged and the motherboard is working. In this case the components most likely damaged are the video card and memory.
You can identify the video card because it connects to the monitor. It's held by a single screw at the top and a clip underneath, near the socket. The memory modules are long, narrow printed circuit boards with rows of identical chips. They're held by clips at both ends. You can take out both kinds of cards and get exact or compatible replacements at a computer supply store.
It makes sense to replace the power supply and memory cards because they're inexpensive. The video card is more costly and is usually worth exchanging only if the computer seems to be working but there's no picture on the monitor. If changing these items doesn't fix the problem, the damage is probably on the motherboard.
Changing the motherboard is a major undertaking that's normally left to qualified specialists or experienced do-it-yourselfers. If you have backed up your work, getting a new computer may be a wiser option because repairs will be almost as expensive. Even if you need files from the old computer, you could ask the repair shop to install the old hard drive in a new machine and your data and files will be available to you.