I've never repaired a lite, but I did fix a GB Micro and DS Phat having the same problem (L or R button not responsive.)
What I did was open it up, until you can see the little switchbox where the actual button is (there's a tiny button that clicks on top, and this is what the L or R button is pressing onto, you can't actually get into the "box"), I put a bit of paper towel or tissue under one site and spray it out with contact cleaner. NOT WD-40, but find something that is made for cleaning electronic contacts, Radio Shack or whatever should sell something that is safe to use. After you spray a bit in, press the button a bunch of times, and let the "liquid" seep out the other side into the tissue. Maybe do it 3 or 4 more times, and then your button should "come back to life". As in, it'll get more and more responsive as the cleaner breaks down junk that is preventing the button from working.
Okay, so maybe I don't really know the inner workings of HOW it works... but I can say that it has worked perfectly on 2 occasions, so no harm in trying.