Most people use their leftover pickle brine to…brine again. So you can throw almost any sort of vegetable into the brine–carrots, shredded carrots, cucumbers again, radishes, ginger, garlic cloves etc.
The simplest dressing is to use the pickle brine, which is vinegar, as the vinegar to a simple olive oil-vinegar concoction. I like to add a little more red wine vinegar to the mix to give it more pizzazz, along with shallot and/or minced garlic. Some people also just mix their pickle brine with dijon mustard and call it a day.
Marinade: you can make a sort of poor-man’s German potato salad by marinading the potatoes in it and mixing with a little mustard and mayonnaise; liven up tuna or egg salad; add a little oompf to braised spare ribs; even on potato chips with hot sauce…
Add-ins: tang up a Bloody Mary; takes some of the “bottledness” out of bottled BBQ sauce; in your refried beans…
And if you really want to go to town, try John Thorne’s recipe for Polish Pickle Soup.