I thought it would be just the matter of finding an adapter but it's actually a lot harder than it sounds.
I've tried myself to salvage lcd screens from dead laptops and since met many who've entertained the idea but in every case, we gave up after researching the matter.
The first problem is that while the connectivity on the lcd screen itself should be standard, the connectics on the laptop's board are completely proprietary. If you're lucky, you will be able to find documentations for the wiring and power supply requirements but if you can't do that you're stuck making your own ghetto plug. As for the backlighting, I don't think the cathode tubes are ballasted so you should be able to fire it up with a cheap switching power supply.
After that, the long story is you'll need to adapt the tmds signal from your dvi port into a lvds one (something like
this that supports the native resolution) and there's no cheap consumer grade hardware that'll do that that I know of. To accomplish that, I've heard of people trying to salvage the chip on their laptop, modifying a graphic card to feed the signal directly into a controller, buy expensive custom lcd controllers with their own adc and vga input and all sorts of ghetto solutions which ended up costing just about in the same ballpark as a decent 1080 lcd screen sells for nowadays.
It's really a shame because there are some beautiful 15 inch 1600x1200 and 1920x1200 lcds sitting on dead laptops that could make this worthwhile but honestly, in the case of a small dvd player screen I would just forget it.