Can I use the leftover pickling brine?
Not sure any of the above have read your question completely.  You are asking; can you store extra brine that you made but was not used in your pickling process.  I do this systematically when my  Green Beans are going crazy.  I mix up extra brine and store it in the refrigerator for up to 2 weeks.  The brine is simple pickling salt, water and white vinegar.  Then I can use it incrementally as my produce comes in. Saves me a few steps by simply reheating premixed brine instead of making it each time.  I like this approach because it allows me to put up a few pints at a time while the beans are fresh.  Have never had a problem with my Dilly Beans.