I'm going to have two main characters who will be a sort of contrast, or foil I guess, to each other. The one I'm worried about is going to be an example of 'Good is not nice', she's going to be rude, sarcastic and unpleasant, but she's also going to be loyal and do all she can to help those she cares about. I'm not sure how to pull this off without alienating her from the reader, or minimizing her snark though. Any ideas?
I'd probably give her a girly or cute secret. A guilty pleasure that she tries to keep secret. Like maybe she likes ice cream with sprinkles. Loves puppies and kittens. Your characters are probably not surrounded by such things so the reader may never know until they come across it rarely to her embarrassment.
You could maybe have her have a soft spot for young children, something substantial like that to soften her somehow. Maybe write some her dialogue witty with an edge and a bit of humor, so the reader doesn't totally hate her.
I don't think there's any one method to do this that we can tell you. You'll even succeed in doing it or you won't. Trial and error. I think the biggest point to remember is that you should treat this character as a real person. Don't just make her entire character these obnoxious traits; make it simply part of her. And heck, if you validate her attitude, readers might even like her. I know TV is different, but often, the smartass characters are the favorites.
I'd suggest that she have a certain impatience with stupidity and hypocrisy, and when she senses it, she tends to be merciless. As long as the "meanness" is guided by some inner guiding principle the reader can identify with, she can get away with an awful lot. If she's being unfairly mean, or mean just for the sake of being mean, then that is harder to empathize with, unless there is some deep and profound reason for the meanness.
I agree with marktx: it's not about giving her a "secret" for the reader to think she's soft inside. I think the matter here is more about explaining why she's so mean, and make that reason something the reader can relate to, or understand. Think of it as a trial of a murder: she's really comitted (commited?) the crime. Does she have a reason good enough to be considered innocent in the eyes of a judge? Same thing with her behavior.
Check out Malcolm Reynolds from Firefly/Serenity. He is the perfect archetype for that type of character. He's sarcastic, grumpy, and mean, but when his crew is in danger, he'd die for any of them. They are his family. It doesn't mean he doesn't rag on Inara, or boss around Jayne, punch Simon in the face, or crack wise ass jokes about Wash. But he'd fight til his last breath for any of them. I'd watch the one and only season of Firefly, and take some notes about his character. Also, I would point you towards Dean Winchester from Supernatural. Neither of them are female, but they are good archetypes for being sarcastic and unpleasant, but keeping the funny and loyalty alive for the viewer/reader.
Another good example I just recently read is The Changelings by Elle Casey. The MC is a teenage girl that tends to be more sarcastic and rough (although not mean, per se), while still maintaining her "good guy/girl" credentials throughout the story.
As a few people have said before, it's more about giving her a personality people can sympathize with, rather than a hobby. Sure, she comes off as cynical, but does that mean she doesn't care for her fellow team members? Of course not! In fact, her sarcasm might be her way of showing people that she actually does care about them. Show her personality in a good light - focus more on the positive, and the reader will focus more on the positive as well.
When it comes to creating a mean or cynical character that you want your readers to like there is no one "trick" that will help with this. There are a few things you could do though. The best idea is to give her likeable traits under the surface. Outwardly, she seems like this mean or sarcastic character, but on the inside she has reasons for acting this way-- say a dark past, or a secret to protect soeone, or maybe a fear of falling in love. Another way to do this would be to let her grow on the other characters. Start her off as cold and unfeeling, but when the team really needs her, she proves why she's useful by being a valuable team player, maybe even risking her own neck for someone else or seeming to care. Then follow it up with a sly or snarky remark before its back to the same od character. At least this way the reader knows theres a softer side. Beyond this, you could give her a soft spot for children, or animals, or nature. Altogether, there are planty of ways to accomplish this, but it all has to be developed slowly as the story progresses. give the reader this hard character and then show glimpses of what is really under her skin
I think that you can create a character that the reader hates but still wants to follow, because they want to see the person get their just desserts. From what i've read though, you seem to want her to have a tough exterior but a soft inside. You can show that her dialogue supports her "badassery" but when she's alone she can be soft, or you can show body language which contradicts her persona.
