This could be a number of things. The logic board could have an issue causing anomaly in the screen. The LCD wouldn't be the problem here because when it starts to age pixels die not lines in the screen.
You could take the device apart and clean it. Try making sure all the connections are not loose.
Here is the guide to disassembling it:
http://www.ifixit.com/Guide/Repair/Installing-iPod-Touch-1st-Generation-Display/634/1
If your not up to that take it to apple and see if they can help.
Regards,
-Varsity