I use cotton balls. It usually only takes half of one or less and they take a spark very well.
- if you coat them with vaseline they will tolerate more water exposure
- vaseline also lets them burn like a candle, making a flame for up to two minutes versus thirty seconds for a dry cotton ball
- neosporin instead of vaseline make them more useful
For a spark I like a ferro rod, or you can use a lighter, the striker from a lighter, etc.
As long as you start with small enough wood, it doesn't matter how wet it is. How wet the weather is, not the wood
I built a fire last weekend just to prove to myself I still could. This was during the rain with wood picked up off the ground. Just start with twigs that are really, really small. You can split wood, also, either with a knife or just with your hands. If you can't break it with your hands, it's either too big or too wet. Dry split wood will light faster but unless you're fighting hypothermia and every minute counts, I wouldn't go to the trouble. In my little experiment it took an hour to tie up some shelter, build the fire, and then boil some water in a can from the ditch.
The cotton balls work so well I only use matches and lighters for lighting candles in the house. I use real charcoal, not briquettes, when I grill, and it also lights very well with a handful of twigs and half a cotton ball.