Strange laptop LCD failure
This is a particularly intriguing problem.
My guess, initially, is that the display controller specifically for the internal laptop display somehow has messed up the way it clocks data out to the actual physical display. My guess is that it probably handles requests a vertical column at a time, and perhaps the timing got messed up. I would think it's a hardware failure.
However, I cannot be certain of this with the limited information I have right now. I'll see if I can research this some more and come back with some more details...
EDIT
I did a lot of googling (well, Bing-ing :P) and it led me to start looking at the tech specs for the display panel itself. What's particularly interesting, I think, is on page 17 and 18 of this document, which is the tech sheet for your particular panel: http://www.displayalliance.com/storage/B140XW03_V_0.pdf
I learned while looking up this information that some panels do have a specific pin on the connector that can specify R/L scan direction. Yours does not, but it does say that horizontal sync is synchronized to RxCLKIN, which is on pins 17 and 18. I'm not very familiar with LVDS interfaces and how this protocol operates, so my attempts to gleam more detail out of what's there won't yield much. However, my guess is that you may have a damaged pin. I'd start with the RxCLKIN pins and see if there's anything that isn't connecting properly, if you want to try and fix it. If you're not comfortable with doing so, I'd say just buy a replacement and hope that it works.
If it doesn't, then the laptop may be messing up the signals going into the display, and if that's the case, you may have to look into replacing the motherboard of the laptop. Or just buy a new laptop. :P Either way, it's up to you at that point.
For future reference, I ended up replacing the LCD panel. The computer/display works fine now, as it used to.
Maybe one day I'll have more time/knowledge/equipment to investigate what happened exactly to the faulty LCD panel. Meanwhile, I hope this question can help others encountering the same issue...
Many thanks to all who helped and shared their knowledge!!!