Hmm... Curious. I can think of three possibilities, but I have never seen this kind of thing before.

1) fuse. Perhaps a blown, or borderline, fuse in the display circuitry? I would have expected significantly worse display results if this was the case, but it might be possible.

2) cable. Check your display cable (from the main board to the display module) to see if there is any visible fraying or damage.

3) software. Perhaps reflashing the kernel, or verifying your software installation. Maybe something is causing problems with your display driver, or something is writing where it shouldn't. Do you have a stock software load, or (if not) what utitilies do you have installed?

Does this line appear when the unit is booting also? Or syncing? That would help tell if it is a hardware problem (especially if it happens while booting) or something in your software.