It’s fun to find things like this! I love cooking with cast iron skillets. Cast iron holds heat well and distributes it evenly, making it an excellent cooking surface. And like you mention, it’s also perfect for cooking over a campfire. And once seasoned, cast iron pans develop a non-stick surface, making cleanup a breeze.
If your grandmother used the skillet, then I assume it has already been seasoned. In this case, in order to remove the rust, there is nothing wrong with using hot water, dish soap, and a scrub brush to clean it. Tough spots can be scoured with steel wool. Dry it thoroughly to prevent rusting again. Once dry, use a paper towel to spread vegetable oil around the bottom and sides of the skillet. Rub the oil in, do not let it pool. It will almost look dry again, just oily.
Now you want to re-season the skillet. Place the skillet in the oven and turn the oven on to 350°. Once the oven gets to temperature, set the timer for one hour. After one hour, turn the oven off but leave the skillet in the oven for another 2 hours. After 2 hours, the skillet should be cool enough to handle. Depending on the condition of your skillet, you may want to repeat this process another 2 or 3 times.
In the future, when you wash your skillet, skip the dish soap and just scrub it clean with a brush and water (if necessary). Dry the skillet thoroughly and apply vegetable oil with a paper towel. Store the skillet like this and it will be good to go the next time you need it.