You can use the same recipe but swap the mayo with brine — start with a spoonful of brine, then add more to taste. If you need mayo, add a little bit of both.
You can also make a vinaigrette with lemon juice and mustard and add the brine, or replace the sweet pickle relish with chopped pickles and pickle brine.
Whatever pickled foods you have are in brine, so don't limit yourself to olive and pickle brine. For some spice, consider brine from hot cherry peppers or pickled jalapeños.