I wish I could give you more information, but here's what I have seen in the laptops I have ripped open:
Usually the LCD itself is no different than any other LCD panel you would pick up at, say, SparkFun (aside from the obvious size and resolution). This in and of itself isn't much, because there are many different interfaces and you would have to identify the panel itself and hope you could weasel something out of the manufacturer. If so, you could probably build something to handle the interfacing, but it would not be an easy task. It's probably right in the wheelhouse of an FPGA, but now I'm really speculating just to give a sense of project scale.
That being said, the panel usually has a ribbon cable which interfaces with some part of the laptop proper, and in some cases it appears as if there is a daughter board that handles the work between the CPU/GPU and the panel. If this is the case, you may find that you can hack together some kind of interface on the far side of the daughter board from the LCD. It seems unlikely but possible, and you'd still be stuck with the task of identifying the interface that is actually used (DVI sounds like a good place to start on a newer laptop) and you'd still have to create a board from scratch to interface with a "normal" video cable.
Even after you figure out the interface, you'd still need to address power to both the LCD and to the backlight. If you can solve the interface issues, these should be a walk in the park. However, it's something to consider since you'll be powering some interface circuit anyway.
Again, my knowledge on this is limited to the few models of laptops I've had in pieces, and I've never looked that closely at the video. Someone else may be able to give you more details and there may be some easier solutions than I've suggested.
I did some poking around on the Internet, because your question really got me thinking. I don't have any more real knowledge than I did a few hours back, but I did run across this site which talks about different LCD types and how you can build custom interfaces with them. It was last updated 3 January 2008 at the time of this post and I have no first-hand experience with this, I'm linking because it describes the lengths someone went through to make a certain subset of panels work.