For tires, I carry an ARB tire plug kit (Speedy Seal), also known as Safety Seal kit. The ARB kit has sturdier tools. It's not cheap, around $40-45, but worth it if you are going out multiple times. It uses a lubricant and self-vulcanizing plugs, as opposed to adhesive. I've never had one fail, and sometimes we've put in 4-6 plugs to repair a cut. As always, when you return home, you should get the tire properly repaired from the inside with a patch. I have a compressor built into my winch, though those aren't sold anymore. Almost any compressor will work; it just depends on how long it takes to air up the tire.
Everything else Joe says is also important, especially the water.A 2x2 foot piece of 3/4" plywood is a great base for soft sand. I also carry a bottle jack, as for tire changing, I hate hi-lift jacks. In additional to Joe's list, I have some emergency food rations, and a big first aid kit.
In thinking further, I would strongly NOT recommend most any rental vehicle on Lippincott. Going up or down, you really need low range, and 95% of rental SUVs will be two wheel drive, and certainly won't have low range. Going up without the lower gears could burn out an auto trans; going down could burn out the brakes.
David Bricker / SYR