PaulS:
The real question is what to do with the data after it is fetched from the serial port. Processing or C++ or C# or VB or any of a host of other tools can be used to read and parse the serial data. When the application knows that "call carol" came in, what then?
You'd have to write a program to drive Skype via its COM API. I haven't done it, but it looks straight forward.
I would have thought that using Gobetwino to start your 'skype launcher, would be simple enough. If you design your Arduino+sketch to emulate a keyboard, you might even be able to do it directly using a windows hotkey, which would save you needing to explicitly deal with sleep issues (the computer would wake on input).
Is it out of the question to have your friend actually use the computer directly? That would make it possible for her to receive video calls as well as make them. If the PC is set up right this can be made quite simple.
If you can budget it, I highly recommend using an iPad for this - no doubt there are other tablet PCs that would do just as well. The sound quality and recording sensitivity blows a normal PC into the weeds, and since you're much closer to the screen and camera you can also see and be seen far better.