Your first idea for coping with offbeat openings might be to try to study all those offbeat openings that you're afraid you'll run into. Maybe you'll get some ideas about how to deal with offbeat openings generally, but don't expect to be able to get prepared for everything you might face. The number of offbeat openings is virtually limitless so clearly this approach is impractical. Also, how much time can you afford to spend studying an opening that you might see at most once in your lifetime? There's just not much improvement bang for your study buck with this approach.
For another possible approach, consider Shereshevsky's advice for dealing with openings "that can not even aspire for the mark of quality". In his excellent book, The Soviet Chess Conveyor, Shereshevsky recommends players not try to battle slightly offbeat openings with the responses recommended by theory. Instead, he recommends preparing a kind of "counter-surprise": prepare a playable but unusual response to the unusual opening. The idea is that a player plays an offbeat opening primarily for its surprise value, not to try to get an opening advantage. By answering his unusual opening with something slightly unusual of your own, you've robbed him of the only trump he has - surprise. Shereshevsky's recommended responses to the Budapest Gambit, the Tchigorin Defence, and the Albin Counter Gambit are good, solid, sensible moves. I think the recommendations would work well, and I think this is a good approach, as far as it goes.
Unfortunately, Shereshevsky's approach has a limitation I've already mentioned. It's just not practical to have something prepared for every offbeat opening you might have to face.
Another common approach can be illustrated using the Sicilian Defence as an example. The Sicilian is the most popular chess opening for good reason. It's sound and aggressive, offering Black excellent chances, not only to equalize, but also to play for a win. It's no surprise therefore that many players study the Najdorf, the Sveshnikov, or the Dragon and anxiously await the chance to bash White. Only White doesn't let that happen. After 1.e4 c5 he plays 2.c3 or 2.Nc3 or 2.f4 or 2.b3 or something else besides the main lines. Black gets so frustrated having to worry about all these sidelines and not getting a chance at his speciality that he dumps the Sicilian from his repertoire.
I think this is a gross over-reaction. If you like the open Sicilian, don't throw the baby out with the bath water. The vast majority of strong players agree that 1.e4 2.Nf3 3.d4 is the strongest way for White to combat the Sicilian. If he plays something else on move 2 or move 3, he's usually either afraid of theory or of the variation you might play against him. Try not to get frustrated. After all, your opponent is playing second or third best moves. Logically, that should be a plus for you.
My first recommendation for fighting offbeat openings has its origin many years ago, when a friend of mine was fretting over what to play against 1.b4. It appeared that in an upcoming round he was going to have to face a 1.b4 specialist who was scoring well with it. The more my friend looked at the book lines against it, the less he liked it from the black side.
I suggested he play the King's Indian, based on the following reasoning. The King's Indian is a respectable opening against the more common closed openings. It seems like it has to be at least as good against 1.b4, which may turn out to be only marginally useful against a King's Indian setup. Besides that, why should you care what books recommend you play against 1.b4? Once you find a response that is good and that you have confidence in, you're in good shape. I don't remember if my friend had to face 1.b4 that day, but at least he was no longer psyched out about facing an offbeat opening.
What are the characteristics of the King's Indian that make it a good antidote to many offbeat set-ups? It's sharp and aggressive, and pieces get developed fairly quickly, but I think that a subtle, often overlooked, characteristic is that the opening moves of the King's Indian are not just reactions to what White plays. If White wastes time or starts playing offbeat moves, Black can keep playing normal King's Indian moves and usually get a better position than if White had followed the main lines.
The Tarrasch Defence to the Queen's Gambit Declined is another opening that is similarly versatile. Black gets to develops his pawns and pieces classically and effectively against virtually any closed set-up by White.
Now consider an opening that is more reactive than the King's Indian and the Tarrasch Defence - the Hedgehog versus the English Opening. Just as in the King's Indian and the Tarrasch, piece placement and the opening moves follow a pattern that's typical of the Hedgehog opening. However, the Hedgehog's characteristic pawn structure occurs only after d4, cxd4. As long as White refrains from playing d4, there is no Hedgehog opening. The opening requires that Black react in fundamentally different ways depending on whether White breaks with d4.
Try the following test in each opening in your repertoire. At each move, ask yourself: If I had one or more free moves here, what would I do? If your answer is "I don't know", then that opening or the way you play it is too reactive. It's also a warning sign that your opening play is relying too much on your opponent following book.
Recommendation 1
Choose and study your repertoire openings so you have a good idea what to play if you have one or more free moves.If you know how to improve your position if given a free move, you're in good shape to improve your position if your opponent plays an offbeat move.
For my next recommendation I'm going to ask you to do a little soul searching. Answer the following question as honestly as you can.
"Why do offbeat openings annoy me so much?"
Logically, they shouldn't bother you. Offbeat openings usually include second or third best moves. That should be good for you.
Strong players don't seem to be bothered much by offbeat openings. Pick your favourite offbeat opening and check your database for what GMs play against it. When I do that I don't see the GMs following an antidote recommended in a book. Instead I see the GMs apparently improvising over the board, finding a variety of ways to beat that opening. They don't appear to have prepared anything special. Instead, they seem to be saying "Good. I get to play against a move that isn't the strongest."
When I get annoyed at having to face an offbeat opening, it's usually because I'm lazy. It's so easy to toss out a few more book moves without thinking. If my opponent plays something offbeat I have to start thinking now.
Clearly that's the wrong mind set. The sooner you start thinking, the more likely you'll win that game. If your opponent is throwing out offbeat moves, you're being given an extra opportunity to get a good position.
Recommendation 2
Adopt the mind set that you'll start thinking at move 1 and that facing the offbeat is an opportunity, not an annoyance.Many of you are probably thinking, "If facing offbeat openings is so tricky, maybe I should start playing offbeat openings myself."
True, you can pose your opponents challenges with offbeat openings but I think you're hurting yourself more than them. Repertoires based on surprise quickly give diminishing returns. You'll either have to keep jumping from one offbeat opening to another to keep the surprises coming, or you'll have to bite the bullet and start building a respectable repertoire you can play long term.
Playing more traditional openings lets you can stand on the shoulders of giants. You're playing proven openings played by the best players. You don't need to rely on following the latest opening theory. Since you're going to be playing the opening for awhile you'll have time to learn the underlying ideas of the opening, which will help you navigate even if something new gets thrown at you. You'll be able to devote more time to other aspects of your game, such as tactics or endings. That's a lot better than constantly having to learn new offbeat openings from scratch.
If you have the right mind set, you shouldn't be worried about facing offbeat openings, and you probably shouldn't be considering playing them. The correct mind set is to try to play the best moves yourself, and to be look forward to those opportunities when your opponent doesn't play the best moves.