Oh, and a couple others, both starring girls: Julie of the Wolves and The Trouble with Tuck. If the kids like dogs or the outdoors, I absolutely ate up anything by Jim Kjelgaard at that age.
ETA: Cheers for The Great Brain 
Other dog or horse books: Bob, Son of Battle, anything by Albert Payson Terhune, and Algonquin, Story of a Great Dog, as well as Will James with Smokey.
Oh! Oh! Owls in the Family and The Dog Who Wouldn’t Be!
Oh, such memories!
If the niece is into horses, Marguerite Henry was one of my favorite authors when I was a girl.
Oldies, but real gems:
The Merry Adventures of Robin Hood, Howard Pyle
Tarzan of The Apes, Edgar Rice Burroughs
I also third the Oz books, or at least the first one that was actually written by Baum. My grandson is reading it (age 11) and loves it.
Can you recommend a specific one to start with? My nine-year-old niece is into horses.
I can’t believe that no-one here has mentioned the great and mighty Garth Nix!
The Old Kingdom trilogy - Sabriel, Lirael and Abhorsen
The Keys to the Kingdom series - Mister Monday, Grim Tuesday, Drowned Wednesday, Sir Thursday and Lady Friday (and they can have the fun of waiting for the final two books to be published too!)
I heartily agree with who suggested the Redwall series, they were great (especially the first three).
There’s also the black magician trilogy by Trudi Canavan - Magicians guild, Novice and High Lord.
And of course the wonderful Dark is Rising series by Susan Cooper, I loved these as a child and they stand up as an adult too.
Black Beauty would be my first recommendation.
To clarify, the first 14 Oz books (of which the first, The Wonderful Wizard of Oz, is the most famous) were all written by Baum. After his death, other Oz books, both “official” and unofficial, were written by other writers—for more than you ever wanted to know, see Wikipedia’s list of Oz books.
I read and enjoyed all the Baum Oz books as a kid, but the library didn’t have the non-Baum Oz books (nor the non-Oz Baum books), so I can’t speak to their quality.
I was an Oz freak when I was a kid. It may be heresy to say this, but I thought the Oz books written by Ruth Plumly Thompson were better than many of the Baum books.
I think I’d start with Misty of Chincoteague.
Oooh, there’s a boxed set on Amazon for a really nice price with the three Chincoteague books I read: http://www.amazon.com/Marguerite-Henrys-Horseshoe-Library-Box/dp/0689716249/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1196285165&sr=8-1
I adore these books, but I would have been terrified by them when I was 10 and under.
Kids have to learn about Necromancy sometime, jsgoddess.
Walter Farley! The Black Stallion, The Black Stallion Returns; The Black Stallion’s Filly; etc.
And my favorite horsey book by Marguerite Henry was Justin Morgan Had A Horse.
I think The Keys to the Kingdom series is for younger readers.
John Bellairs scared the living shit out of me when I was 9-11. I still find his books damn creepy. I spent a whole summer avoiding the bookshelf that had The Lamp from the Warlock’s Tomb on it. Really. I can still see scenes from it and The Curse of the Blue Figurine in my head. Does anyone else remember the bit about the kid walking home at night, hearing leaves blowing, and realising it’s a death mask blowing towards him across the ground?
Aagh!
Anyway, if they like to be scared, he’s a good author.
For the girl, I suggest almost anything by Elizabeth George Speare, if she’s remontely interested in history. The Witch of Blackbird Pond (about Puritan New England) and Calico Captive are really good. I second The Egypt Game,, too. Very good read.
The LHotP series for both. The boys might like Farmer Boy the best.
For the boys, The Neverending Story, by Michael Ende. And Momo.
A couple more books that haven’t yet been mentioned, at least not that I noticed.
The Enchanted Forest Chronicles, starting with Speaking with Dragons by Patricia C. Wrede.
Not precisely scary, but dealing with ghosts, are two of Richard Peck’s Blossom Culp books: The Ghost Belonged to Me, and Ghosts I Have Been.
For a pre-teen boy with a taste for sea stories, or the macabre (amazing how those two go together) I’d also reccomend any of the books of Edward Rowe Snow. I’m not sure how many are still in print (if any) but they each contain several accounts of ahem interesting accounts of nautical lore. I found them very readable, and a lot of fun as a pre-teen. But, they have frank accounts of things that people do to survive shipwrecks, or how deprivations of long voyages can work on a person’s mind. So, if you want to avoid letting these children being confronted with the reality of cannibalism, for example, you might want to avoid these books.
Oh, on a reverse note - STAY AWAY FROM ERAGON! Friends don’t let friends read Christopher Paolini.
Duly noted. NajaHusband wanted to buy them this year for the older nephew, but I beat him into submission with my stuffed Smaug and got the kiddo the My Side of the Mountain trilogy.
Is it really that bad, for a kid? I mean, I’m not interested in reading it myself, but my 11-yr-old son just started it. Several of his friends have read it.
He’s gone through Harry Potter and Narnia and Lemony Snicket and Roald Dahl and several others listed in this thread. Loved The Hobbit, but couldn’t get into LOTR yet. I’ll get him to try the Prydain books next, I think, although I haven’t read them.
And thanks for the horsey book suggestions. I’ll get my niece the Chincoteague books, and Black Beauty if she doesn’t already have it.
For horsey books, The Black Stallion is great–I think I must have read it a hundred times–but I loved Farley’s other horse books, too: The Island Stallion and Man o’ War and such.
For a horsey book that she may not have read, try Will James’ Smokey The Cow Horse and My Friend Flicka and Marguerite Henry’s King of the Wind, which I read into tatters, two copies. As far as Marguerite Henry goes, I liked Brighty of the Grand Canyon and really loved Justin Morgan Had a Horse, but had little use for the Chincoteague pony books–go figure.
PS: Do not get her Steinbeck’s The Red Pony with intentions of a horsey kid story in mind. Turns out I liked Steinbeck and went on to read and love his work, but damn.
For anyone who might not have read it:
The pony dies and the kid sees vultures picking out its eyes. Quite unexpected for a nine year old.