Iodine, bleach, a fairly comprehensive first-aid kit including oral and topical antibiotics, aspirin, and prednisone and benadryl. A tent, a survival blanket, a firestarter kit (I like the flint and iron strikers we used in colonial kitchens when people were around), matches, rope (nylon and hemp both), needles, thread, SOAP (lots and lots and lots of soap) insect repellent, insect killer, a tarp, several rolls of duct tape (because it is ALWAYS useful), toilet paper (lots and lots of it), a good knife and whetstone (or a few different ones), compound bow and broadhead tips, a handgun and several boxes of soft-tip ammo (handgun because I'm a better shot with that than a long gun, soft-tip to maximize the damage it does on expansion), solar charger of some kind, hand axe, saw, multihead driver, mallet, crowbar. And sunblock. And a big floppy hat. A book of local plants and animals for identification. For entertainment, I'd want lots of books. Anthologies are good as they're books within books. And my dogs. Horse would be helpful (though he's not exactly a work horse.) Something to write on.
Well, I will admit, when I saw the movie version, my first thought was "Those are the wrong kind of arrows they gave Katniss in the arena" (vertically-turned broadhead is meant for targets with vertical ribs, while what she was shooting at, uh...well, the human rib cage is horizontal...) Of course all the hunting and fishing equipment and guns and bows and nets won't help if there's nothing to hunt on the island...as far as plants go, though, I don't eat it unless I am SURE I know what it is so a guidebook for the area is a must!