If you blow a tire, Murphy's law says that it will be the right side tire and it will blow out at night on the darkest stretch of road on your trip, in the rain. So, have a spare tire. 

I have a small plastic tub that is the very last thing that I either put in the trunk or lash to a convenient spot on the trailer after it's loaded. It has in it a scissors jack and handle. A square of plywood or a board to put under the jack, especially if the ground is soft. A breaker bar or rachet with the correct size socket, and a couple of proper sized lug nuts or wheel studs, depending on your trailer. (ask me  how I know). A small plastic tarp, an extra ratchet strap or two is never a bad idea, as are a few extra bungees. 

I also carry a headband light, and a light I can stand on the ground, as well as a gardeners kneeling pad. Kneeling on 100°+ pavement, gravel, or in the snow, slush or mud is no fun. Those 18 wheelers whizzing by in the rain throw off a hell of a spray. Have a couple of cheap ponchos, some gloves, and a towel. Some of my buddies with boats carry a can of fix-a-flat.
 
A couple of notes about the jack...can you use your car's jack? Probably. But...take it out of its storage spot in the car before you go. Chances are, you're going to have the trunk packed full for the trip and you don't want to have to unload the trunk on the side of the road (in the rain, remember Murphy?) to get to the jack.
Your car jack handle isn't very long because you don't have to get the jack very far under the car. It's a longer reach to get from behind the tire to under the trailer axle. That pivot/twist/folding handle doesn't really extend far enough to operate the jack easily. Find a jack with a long handle.

Set a wheel chocks so if you have to unhook from it it don't go rolling off