I would just bring something small with you, like your favorite candy that you can share with them the first night or something. Then I would get to know the family better and buy them a gift before you leave, this way you can get them something you know they will like and use. Or maybe there will be something that you have that you want to leave behind with them, for instance we gave our family one of our DVDs because our host sister watched it over and over again while we were there.
A raincoat, it was really hot to wear it and Jamaicans don't use them so I really felt like I stood out and looked silly. I brought it back to the States the first time I returned. A nice sturdy umbrella that you can fit in your bag and carry all the time is really a much better option.
I've had tons of packages sent to me which has been really awesome. Here are a few of the things I have requested: towels (both quick dry and regular bath towels), a solar shower (really great when you have to take a bucket bath because you can get a better stream of water), a coffee grinder and french press, a good quality can opener, and a good quality kitchen knife.
One main difference is the fact that you can crank the one to recharge it, so it would be better in a case when you had no electricity and you ran out of battery power. The other advantage of the hand crank radio is that many have a cell phone adapter so you can recharge your cell phone as well. I have both and the primary reason I got the crank one was for the cell phone adapter. Also the main advantage of the world band is that it has shortwave radio, but Jamaica gets tons of FM stations (including BBC) so you don't ever need the shortwave stations. I'd say go with the emergency crank kind.
We have sleeping pads and find them pretty handy because we go visit other volunteers a lot and have people stay at our place pretty often. All the floors in Jamaica are concrete covered with tile so having that extra cushion is really nice. I don't really find that a sleeping bag is too necessary, we usually just bring a sheet because it is easier to carry and most places aren't very cold.
We checked our backpacks as they were and just buckled and tied up all the straps and they got through fine. But there really isn't any advantage to having a backpack because you won't be able to travel very easily around the island with a backpacking backpack (and it makes you stand out which could make you a target for crime) so I would recommend bringing 2 duffel bags or suitcases for your checked bags because they will be easier to pack.