I've seen a few solutions to this which I"m struggling to get going - it would be nice it it was build into the Arduino interface.
There are other serial monitor applications that have this ability. I don't think most users would need this.
The primary purpose of the serial monitor is to debug sketches, not to be a data collector. If you need a data collector, you are using the wrong tool.
Hyperterminal has this capability and shouldn't be to hard to get, if it didn't come already installed.
If not - You can Highlight Text in the Monitor window and Paste it into Wordpad and save it from there.