I'll third b_b. The pickletini is actually quite delicious, but only if the pickle brine is of exceptional quality (i.e., salty, but not obscenely salty, and teeming with various spices, herbs, and macerated extras, such as cloves of garlic, (nosemuffs, thenewgreen!) wedges of onion, whole habanero peppers, lemon rind, etc.) and used somewhat sparingly, so that the subtler qualities of your gin/vodka of choice will still be able to make an appearance. Whether to include or not to include vermouth will be a hotly contested point. Between the expansive styles of a fine pickle brine and a quality vermouth, your cocktail glass may not be big enough for the both of 'em.
caveat imbiber: made in anything other than perfect proportions, this cocktail is either a. undrinkable or b., pretty much indistinguishable from an ordinary dry martini. Same goes for all dirty martinis, though, in my opinion.
Sadly, beyond bloodys (mary and caesar work equally well) and the pickletini, not much else comes to mind. The scarcity, I attribute to the general scarcity of savory cocktails. This latter scarcity I attribute to the fact that, generally, when craving savor, one says "let's eat!" rather than "let's drink!" Still, it is a somewhat galling lack. Many cocktails dally in savor country (saketinis, sage margaritas, [aka, paparitas], cilantro mojitos, jalapeno margaritas, and so on), but none of these delicate balances between sweet/salt/tart can withstand the briny intensity of a proper pickle juice. Used in small enough proportion to behave, it will disappear.
The one other "drink" which comes to mind is the pickleback. For the uninitiated, this is a shot of whiskey followed by a shot of pickle juice. While I hardly consider this a cocktail in any traditional sense of the word, it is without doubt a singular experience, and one worth having at least once. As bizarre as the pairing sounds, something about it works, and by "works," I mean, "does something in your mouth that has never been done there before." Whether you find this thing epiphanic or disgraceful is impossible to predict beforehand. The combination is singular, and might lead us to experiment further, but the two seem to make sense only in sequence, not in combination.
If we can crack this nut though, we'll all drink more healthfully. Pickle juice is, after all, an amazingly rich conveyor of electrolytes, which, as has been noted, are key to avoiding any ill effects of your revelry. I've been told it was the coconut water of the early twentieth century. Let's see the energy drink industry make an ad campaign out of that one.