14 gauge is actually smaller than 12 gauge. As wire gauge numbers go down, the wires actually get bigger. 12 gauge wires can safely carry more current than 14 gauge, so I would suggest you go with the heavier cord as suggested by the directions.
What are you planning on using the generator for?
What started smoking - the weed eater or the cord? I would guess a bad connection or faulty device rather than an issue with the cord size.
Depending on how much weed eating you have to do, consider a cordless model if it's not a huge lot or a gas model if it is. Dragging a cord around is pain for a big lot and a hassle to take out every time for small lot. However, a gas model might be too finicky for you and your son to deal with.
Sorry to hear that. Best of luck to all 3 of you.