The LPC1768 supports Wake-On-Lan (WOL). It's fairly easy to program, but something will have to provide the "magic-packet" to wake it up at 8AM and your program will have to be coded to notice it's midnight and shut itself off.
I've only used this approach for designs using larger numbers of LPC1768s dedicated to specific functions to bring the required LPC1768s online as needed and power them off when they weren't necessary or wanted.
If your design is simple, e.g. one mbed, an easier way may be to just make a time aware power supply.
If you're powering through the USB port, you could program your PC to disable and enable the USB port the mbed is powered from at the required times (a cron type program). This works with some machines and USB ports, especially notebooks. However, some machines and USB ports provide vampire power unconditionally.
I would like my mbed to automatically power up at 8:00 AM every morning and automatically power down at midnight every day. I have already set the correct time on the mbed, but I wasn't sure if there is a way to automatically power up or down the mbed. Does anyone have any suggestions?