Home Electrical Wiring, Experts Advice Please

[RyanN]

I know this forum isn't for discussing personal home electric problems, but I couldn't find the answer on google and I wasn't sure myself.

Today, an outlet in our kitchen, the light above our kitchen sink, and our dishwasher died, but the GFCI outlet they were connected to was working. My first assumption was that a wire fell off because that's exactly how it seemed until we opened the box. We shut off the breaker, pulled out the outlet, and found a rather scary charred mess. There were wires melted behind the GFCI outlet on, which came out of the load side and it was the hot wire. Whoever wired our house put some small stranded 16AWG jumper wire that went to a wire cap that fed the dishwasher and the other outlet. The GFCI Outlet was rated for 15AMPS and the circuit breaker in the panel is 20AMPS. So my question... My dad wants to reuse the 16AWG wire that is connected to the white wires (since it is not burned), but I'm a bit worried that the same problem may occur with the white wires if we replace the hot black wire with something more hefty (like the solid wire that should have been there). I'm also oblivious to what caused this because the outlet, dishwasher, and light were working earlier today. Also, the wire seems to have been melted for some time, but I'm not sure. There are some signs of corrosion, but not very much, and the remnants of magic smoke still lingered. The only thing connected to the outlet was a coffee maker that has been off all day.

So if you could please answer the question, and if someone could explain why the wire melted, but neither the GFCI outlet nor the circuit breaker tripped. This is a complete mystery to me to how, when, and why this happened as there were not symptoms of a fire in the box.