Why does my DS Lite's top LCD only display white?

Broken ribbon cable, get a new case and top screen.

You don't need to get a new case if it's not broken, you can disassemble the DSL with a Phillips #00 and a TriWing driver. There are two possibilities how the problem could be occurred. First, and that's the one cheaper and easier to fix, the ribbon cable from your top screen is broken and you just need to change the top screen. Or, second, the flap attaching the ribbon cable to your logic board is broken. In that case you have to change the logic board, but these are about 50$, so it would be cheaper and easier to buy a used DSL from eBay etc.. If you want to replace the screen (or the mainboard), maybe it would be an idea to buy a broken DSL with a faultless screen or mainboard, disassemble it and replace the broken parts with the intact ones. Somehow or other, repairing a DS isn't as difficult as it looks like, you just need sure instinct. Give it a try.