My laptop didn't just suddenly stop charging. No, it would charge if I had the cable in JUST the right way... you know how it goes. After I bought a new cable and found the same problem, I knew it had to be the port. Angst and despair set in.
It was a long and scary process. I've replaced the logic board on an iBook before, but it required less work. I found and cleaned out about three cats' worth of fur inside while I was at it, and thought for sure I was going to destroy the microphone detaching it.
Screw down the Magsafe port slightly before putting the motherboard back in. It popped up above the logic board and I had to unscrew the logic board in order to wiggle it back down into place. Get about a dozen little yogurt cups, Tupperwares, or other small containers and keep each step's screws separate.
Phillips #00 Screwdriver
MacBook Unibody (Model No. A1278) MagSafe DC-In Board