It depends on the PC and the settings. Some have wake on lan settings, where if you send an IP address to the machine, it will wake up. But that is probably more complicated with a digispark and overkill.
Some have a setting in the firmware to wake on USB device. It is probably meant for moving the mouse or keyboard, but it may be more general to wake on any USB event. In which case you could have the digispark send a message down the USB serial line. If it is only mouse or keyboard events, then you would have to have some firmware on the digispark that simulates pressing a shift or control key momentarily. In the Arduino world, the Leonardo is more suited for this than the Uno/Mega are. I suspect the digispark may be suited for this as well, but Erik and company would have to chime in.
If you don\'t mind hacking the computer itself, it should be a simple matter to use an opto-isolator to simulate the press of the power button (you would do it in parallel to the real button on the case).
You could use the relay shield to turn power on/off to the computer (feeding the power line through the relay).
If you don\'t want to hack the case, you could always use the digispark to drive a servo to press the button. One of the previous kickstarter projects I backed was \"Roy\" which is a robotic arm/hand/finger and I bought 2 fingers, and I intend to use them to press buttons like camera shutters.
I tend to think using the digispark to simulate a keyboard to do the wake up is perhaps the best solution.