Complement to the last rant.

1) If you have sweet memories of a murdered family, please try to make them sweet and not saccharine.

This happens both with protagonists who escape just as their family is being murdered and those who were orphaned at such an early age that they don’t really remember their parents. The memories they call up to keep them going, give them fuel for vengeance, are told about by other people, or whatever, might as well be seen through rose-colored glasses. The protagonist doesn’t seem to have stepped out of a family. She seems to have stepped out of a Hallmark card.
I suppose someone might argue—I have seen it argued—that the family is there only for symbolic purposes in such a case, so the Hallmark card does no damage. But I don’t like it when other characters are sacrifices to the protagonist’s self-esteem. I don’t like it at all. It’s no different than making one character the designated bully or comic relief. And really, isn’t it boring to write about a human character raised by a family of perfect robots? At least as boring as writing about a human character raised by a family of abusive automatons? Especially when the character herself exhibits traits that aren’t consistent with being raised in such a perfect family, or the daughter of two people who had nothing wrong with them at all?
The best solution, I think, is to show that while the protagonist can remember her wonderfully perfect Momma and Poppa all she likes, that doesn’t mean they never cursed or got drunk or made mistakes or woke up with bedhead and halitosis. Snap the connection between your protagonist’s perceptions and the world around her. Show that she’s not right about everything, and then the readers are more likely to accept her polished portraits as the defensive mechanisms of a grieving person, rather than the HOLY TRUTH as it’s often portrayed in fantasy novels.

2) Don’t ignore serious problems—like clashes of personalities—that the story sets up just to have everyone defer to your protagonist.

Family interactions are an interesting middle drawing in most fantasy novels, and even short stories, between the fully developed, deep, and human picture that there would be if the author ever got off her lazy ass and the scribbled, barely penciled-in sketches that are most interactions between the protagonists and stock fantasy characters. Authors, perhaps drawing on memories of their own families, might start to set up situations where the protagonist isn’t always right. The sibling relationships can seem very real. The parents might have understandable reasons for not wanting their daughter to practice fire magic in the middle of a dry forest when there hasn’t been rain for weeks. The protagonist might make a severe mistake and seem about to get punished with it.
And then what happens?
The author defuses the tension, has the other person forgive the protagonist, and the story goes smoothly along.
I want to know where the authors find all this magical forgiveness. I wish I knew already. I’d buy it by the truckload and make my brother forgive me for every time we’ve annoyed each other since he was nine months old.
The thing is, you can’t have the fruits of conflict without the conflict itself. For pain and mistakes to cause character development, they have to happen in the first place. Rescuing your protagonist every time she seems about to get yelled at, probably by having her recite some sticky saccharine speech that would be right at home at the end of a sitcom and the other person blush and back down, doesn’t allow them to happen. It dumps the magical forgiveness sand on the whole bonfire. There’s the protagonist burning down half the forest and her parents racing towards her, making sure she’s all right, opening their mouths to yell…and then there’s the protagonist and her parents sitting on the couch, holding hands, while they assure her that it’s all right and she must have had good reason to use her magic the way she did, never mind the forest fire that threatened the village and destroyed hundreds of innocent trees, The End.
Remember: This is the plot you created. This is the conflict that you set up. You are doing your readers and your characters a disservice if the conflict then doesn’t happen because no one in the protagonist’s family can stand to get angry at her, but most of all, you’re doing yourself a disservice. This kind of shortcutting encourages authorial laziness like nobody’s business. And it is nobody’s business. If the character never makes a mistake or pays for one, she’s going to remain one-dimensional.

3) Try introducing some slippage between children’s perceptions of famous parents and what those parents did.

I tend to back away from bad fantasy at a cautious pace. When I pick up a series that’s about the children of a world-saving couple from a prior series, my speed approaches the supersonic.
The series serves to let the children become copies of their world-saving parents, or let the parents save the world again, in 90% of the cases I’ve read. (I am so talking to you and your entire plot for the Malloreon, David Eddings). And the children have heard all about their famous parents from the time they could toddle, of course, and though they may at first be resentful and want to escape that shadow, in the end they will reconcile and All Will Be All Right.
You don’t think any world-saving couple would ever try to shelter their children? You think the near end of the world always makes for appropriate bedtime stories? You think that the stories the couple tells their kids about themselves would perfectly match what the old cronies of the couple, or their enemies, remember? There are dozens of ways that the “truth” would get distorted, and the children of the famous parents might well not remember what their parents really look or sound like, from the stories that come to their ears when they start listening. Think of the way that tabloids represent celebrities. Those are assuredly very different from the things that celebrities think about themselves- and tell their children.
So, if you must write about the children of a world-saving couple—and I would say think twice as hard about that as about any other kind of fantasy plot, except perhaps the one with the royal orphaned heir adopted by peasants who has a mysterious prophecy attached to him saying he will defeat the Dark Lord—give them a chance to get caught in that delicious middle ground between rumor and truth. Don’t let their parents be shining beacons. If the first series was good, they probably weren’t shining, flawless beacons then anyway, were they? If you have more to say about them, they shouldn’t have transformed into shining, flawless beacons between series, either. Make them (and their children) people, and the family can get along just fine without perfectly getting along.

4) If children have supernatural intelligence and/or a sense of responsibility, then show a family that would foster those abilities.

This goes straight back to my reading of The Painter Knight by Fiona Patton, in which I had got about a third of the way through the book and started thinking, “This has a kid in it. The kid is five years old and doing stuff that would be exceptional for a child of eighteen…Oh shit oh shit oh shit.”
I did finish the book, but The Painter Knight remains the only book in Patton’s Branion series that I’ve never had the urge to reread. It can stay in its box, thank you.
Why? Because the child in question, five-year-old Prince Kassandra (no, that is not a misprint; Branion is a gender-neutral society) has a father whose air is one of benign neglect, who isn’t a very good ruler himself, who spends a lot of time drinking and carousing and doesn’t teach her how to handle their family line’s inherent flame-based magic—and then, when he’s murdered and she has to flee with her father’s best friend, Kassandra is somehow the most intelligent and best ruler that Branion has ever seen. At five. She judges the cases of traitors and comes up with war strategies. At five. With that kind of father. At five.
I’m not going to say that fantasy children should never ever ever have abilities beyond the ordinary, because a medieval environment could well foster early adulthood, and besides, George R. R. Martin devotees would have earned the right to smack me for saying it. But do place them in a family where parents and/or siblings might have that kind of influence, hmmm? While one could argue that Prince Kassandra’s inherent magic told her what to do, none of the other royal children anywhere else in the Branion series show similar abilities, even in equally dangerous circumstances. She’s Just That Special.
At least show some reason for why the child became Just That Special.

5) Don’t play the “Anything you can do I can do better!” game between the protagonist and her parents, or the protagonist and her siblings.

This goes back to 3 to a large extent. When the famous world-saving couple’s children rebel and run away, they often develop abilities that are carbon copies of their parents’. They’re also going to be the chosen of their parents’ gods, or they’re going to be fire mages, or they’re going to be the best fighters the world has ever seen, or whatever—except that they’ll do it even better. This is the prime reason that a second series focused on the couple’s children so often feels like an excuse for the author to rewrite the same concepts she’s already written once.
It also sometimes happens with siblings. All the children can talk to animals, but the protagonist has the most powerful magic—except that no one knows that, probably, because it’s hidden behind Stupid-Ass Plot Devices. So her siblings tease her, until the climax, when the protagonist talks to all the animals at once and brings all of them to help, and then her siblings collapse at her feet in helpless admiration. Piffle. Psst. Rubbish. But it happens anyway.
This is the obverse of the antagonistic and petty sibling relationship based on jealousy, while sharing some of its worst characteristics. Here, other characters can do things the protagonist can do—but she’s still the best out of all of them. They possess no skill that she does not. She is the container and repository of all magic, or all wisdom, or all skill.
Please, take this shit over there. I think the rosebushes need it more than the story does.

6) If you have twins, try making them separate people.

I don’t say “different,” because it’s perfectly possible to make twins different—often obnoxiously so, so that one twin is wild where the other is refined, one is popular while the other is popular only with thesuckers elite, one is extroverted while the other is introverted—and still not have them be separate people. They’re too obviously foils for each other. The story could have one perfect character, and no one would notice the difference. But because the author senses, dimly, the peril of describing someone as both wild and refined, popular and unpopular, extroverted and introverted, she separates them into two mirror images instead, just so that she can have them all the traits in boring perfection.
I get bored reading about twins because the author tells me over and over again about the “special” bond between them, while not showing it. There’s this thing about a bond, you see. You have to have at least two people to make it. And if you don’t have two people who register as people, instead of one person the author just happens to have two names for or an attempt at including all the possible perfect traits in diametric boringness, everything from Column A and everything from Column B, then I’m not interested.
Try thinking about what there is to say about the twins, besides the fact that they’re twins. If you really can’t think of anything about them that doesn’t depend on a complementary presence or lack in their twin, they’re probably on their way to boring perfection.

7) Don’t upstage another family with your protagonist’s family, and vice versa.

You know the tale:
Once there was a bitter, lonely, unhappy protagonist who suffered at home because his family didn’t appreciate her. Then he went to her best friend’s house, and they treated him well and loved her, and everything was better there. The End.
Once there was a bitter, lonely, unhappy best friend of the protagonist who suffered at home because they didn’t appreciate him. So she went to the protagonist’s house, and got adopted by his wonderful family, and everything was better. The End.
Comparing families ultimately does no more than comparing abilities. It’s almost never a true comparison, darks to lights, advantages to disadvantages, but a tally that puts maybe one pro under the terrible family and gives all the others to the perfect one. And the perfect family always appreciates and loves the protagonist/best friend for who he or she was meant to be, of course. Somehow, they can understand this inherent specialness that other people can have lived with her for ten years and never seen. (Maybe because it only exists in her author’s head?)
One thing I appreciated, for all my irritation with other aspects of the story, about Robin Hobb’s protagonist, the royal bastard, FitzChivalry, is that his family is not ideal (his royal grandfather trains him to become an assassin so he will be loyal to him; his uncles are well-meaning but busy, or vicious; his father’s wife tries with him, but doesn’t always succeed; he doesn’t remember his mother at all), but neither is the family of his best friend and love interest, Molly (a neglectful father, and no adoring siblings). They had to find family in other places than each other’s set of inexplicably perfect blood relatives.
Travel in fantasy is next.