A lot of fantasies take half-elven characters (most often; other half-breeds are rarer) as protagonists. That in itself isn't bad. What annoys me is the way that these mixed-race characters always seem to be portrayed.
1) Any inheritance from the non-human side is always an instinctive advantage. Half-elves tend to be keener of sight and hearing than humans, for example, and have longer lives. Usually they mourn not having lives that are even longer, or more elven magic. There's less ink spent on them outliving humans than there is on them not living as long as elves, and no ink that I've ever seen spent on them learning to deal with their advantages. How would a young half-elf learn to use the keener sight and hearing that no human would be able to provide lessons for? For that matter, how would a half-elf who grew up among elves learn to deal with lesser variations of those some senses? Both societies seem to make the adjustment without ever noticing. Surely a young half-elf, though, would notice if she could see things better than most people, or if everyone else was always listening to sounds she couldn't hear.
In the case of magic, lifespan, and other more notable physical differences, some idea of how the hybrid character manages to adjust would be nice. Is it really instinctive, the way that human babies attempt to learn to walk and speak? And if it is, would that frighten her parents? Or does she have to be taught to use her magic and other advantages, and if so, where does she get the teaching?
2) There is not usually deep prejudice against the half-breed character, but there is whining involved. By "deep prejudice," I mean a serious risk of being killed or injured. I've seen half-elven characters who grow up among humans who hate elves, but somehow the worst they do is bully her once in a while. (Not that the character doesn't use this for a good deal of whining and crying). I would think that any village with a half-elf born into it would have to do a good deal of adjusting. Perhaps they wouldn't be able to adjust, and would drive out the human parent. Or perhaps they would calm down, but only until the half-elven character started showing signs of enemy magic, which would agitate them again.
Either way, the adjustment process seems to take only as long as the author thinks she can wring it for sympathy, and it always involves bullying designed to make the character more sympathetic. The relationship of the half-breed character to the people around her is never that complicated.
3) Don't let the mixing of blood only lead to advantages. As a few people mentioned in the last post, donkeys and horses can breed, but the resulting offspring, mules, are sterile. I don't think I've ever seen a sterile half-elf without huge advantages to compensate for it. (In fact, the only examples I can think of, Lynn Flewelling's half-Aurenfaie, have very powerful magic and privileged positions in the human realm for most of their existence). And sterility is rare in any case.
The mixing of blood will not necessarily produce someone with the advantages of every race, though- and in the case of a character who suffers prejudice and isolation, it really shouldn't. What about characters with a dragon or unicorn parent, ignoring the mechanics for a moment? Somehow they inherit the shapeshifting ability, but never inconvenient tails or horns that get in the way when they try to go through doors. Half-elven characters pity themselves, but are never human in build or even average-looking. And mothers of half-breed characters who die in childbirth do it to provide angst for the character, not because of the difficulties and dangers that would probably be involved in producing a child partially from a different race or species.
Basically, consider ways that cross-breeding might play out to the resulting child's disadvantage as well as advantage.
4) Consider the mechanics. Really. Humans tend to find elves attractive, the epitome of beauty, but what reason does an elf have to romance a human if they're supposedly so much uglier? Do they fall in love easily? Do they feel compelled to honor their one-night flings, if sex is marriage to them? Were they drunk? Was it an experiment? Too often the answer is love, love, love all the time, or rape, and nothing in between.
For species further apart than humans and elves, the mechanics are even more important. How do humans and halflings have sex? What about still-smaller races, as most fairies are portrayed? Usually the answer is "Don't think about it," but the author, if no one else, should be thinking about it.
Finally, a special case comes up with characters who are supposedly half-animal, or half-shapeshifter. The author gets around putting bestiality images into her reader's head by saying that, say, a dragon turns into a human, and has sex with and gives birth in that form, so that the child is born safely but still has dragon heritage. If a dragon has altered itself enough to be able to successfully mate with a human, though, wouldn't its genes be human, too? The child should be born completely human in that case. If the shapeshifting was only an illusion, then how did the dragon and the human have sex in the first place? This is always something I start to wonder about with characters who supposedly have animal heritage, and it leads back, once more, to that idea of giving hybrid characters all the advantages and none of the disadvantages.
5) Don't always make the child raised in the mother's society. If elves have less prejudice against half-elves than humans do, and the father is elven, wouldn't it make sense for the human mother to go to the elves instead of staying behind in a village where they might kill her and the baby? Or perhaps she hands the half-elven child over to the father and goes back to the village. That doesn't seem to be a very popular solution in fantasy, though, however practical it is.
There are ways to get around the child automatically being raised by the mother because the mother was the one to give birth. If the mother is the non-human partner, she could easily be around and up more quickly after the birth than a human woman could. Perhaps the father is with her if she dies of complications. Perhaps she comes from a race that doesn't really care for its children, so she dumps the child on the father's doorstep and goes on with her life. It would all depend on what solution fits the story best.
Remember: to get the half-breed character to an age where he or she can go adventuring, he or she has to survive childhood. And it often seems as if the childhood takes place with a mother who doesn't care and in a society violently prejudiced against the half-breed, where the character should never have survived at all.
6) Consider happy half-breeds occasionally. If the human partner and the non-human partner really were in love, as a lot of fantasy stories insist they are, why would they break apart and go back to their own worlds at the first sign of opposition? Perhaps they manage to make a stand. Perhaps they flee together, taking the baby with them. Perhaps there are separate half-elven or half-dragon villages that accept mixed families. Perhaps they live somewhere far from the cities, where no one cares (it's unlikely that the elven equivalent of a border lord would suffer much opposition if he chose to love a human and raise a half-elven child). Most of the time, these options seem closed off not because of legitimate reasons but to give the half-breed character an opportunity to angst.
There can be plenty of reasons for characters to angst that aren't focused on their race. And if the parents were in a loving relationship, there's no need to make it automatically doomed. I'd like to meet a half-elf, just once, who was raised by other half-elves and therefore doesn't feel compelled to make "the choice!" between the two races. Or one whose parents loved each other, accepted the bond in the full knowledge the human would die first, and raised their child to be strong enough to accept the world's prejudice.
Reading over the list, I noticed that I was always assuming the character was half-human and half something else, instead of, say, half-elf and half-orc. There are probably too many possibilities in that last to list them all, though.