It sounds a fairly simple project, basically just a multi-channel timer, should be easy enough to do.
Obviously you need some kind of switching signal when you plug a unit in, this could be as simple as a plug with a shorting link across it. This simply acts as the 'start button' for the timer on that channel.
Personally I'd probably use multiple PIC's to do the job (but I always use PIC's) - use a small cheap 8 pin PIC on each channel, with a larger PIC doing the display and overall control.
But as long as you've got enough I/O, or don't mind using port expanders, a single micro-controller (of any type) should do it easily.