YES, you can reuse the brine from cultured vegetables
a couple of things before you do.
If you add a starter culture to your cultured vegetables, like Cutting Edge Cultures or another starter culture, then you're going to have extra bacteria in your culture that are stronger than just using salt and no culture. Therefore, you can reuse the brine once you've eaten all the vegetables to make another batch. However, the longer you wait to use the brine, the fewer probiotics you will have since bacteria need food to stay alive. It's best to use the brine within a couple of weeks to make more vegetables and this will ensure you have lots of probiotics in your vegetables. You can use it again and again or until you notice the brine getting super cloudy and perhaps developing some Kahm yeast. This yeast isn't harmful, but it can change the taste of your vegetables. You can also reuse the brine if you used kefir whey instead of a culture package; however, the taste can be a little off-putting because of all the added bacteria in kefir whey.
You should use at least 1/2 cup of brine to start another quart or half-gallon jar of vegetables. (If you have more brine than 1/2 cup you can use this as well.) If you are making a gallon, then a cup (or more) of brine is best. You'll need to add more salt, and add water to cover your vegetables; and then let it ferment again on your counter the same amount of time as the recipe calls for.
It's kind of exciting how versatile these foods can be. You can make lots of cultured vegetables easily and affordably because these microbes do the work for you.