
Amazon | Project Gutenberg
A charming collection of Stevenson's poems for children, on subjects such as bedtime, play, imagination, and the outdoors. The "with images" version of the Project Gutenberg edition includes illustrations by Jessie Wilcox Smith.
Amazon | Project Gutenberg
Little Teddy Platt treasures a button from the uniform of his soldier father who died in battle, and longs to be a soldier himself one day—but with the help of the minister Mr. Upton, he also learns to enlist under the banner of the Lord and fight the battle of spiritual warfare against his own worst enemies within.

Amazon | Project Gutenberg
A classic and beloved girls' novel, Rebecca of Sunnybrook Farm follows the memorable character of Rebecca Rowena Randall, a spirited, curious young girl sent to live with two spinster aunts, through mishaps, adventures and friendships at school and at home as she grows to young womanhood.

Project Gutenberg
Widowed Mrs. Pepper and her five lively children struggle to make ends meet, and eventually enter upon new adventures after making friends with a wealthy old man and his young son.
A free edition is not currently available at Amazon.

Amazon (Translation I) | Project Gutenberg (Translation II)
Orphaned Heidi, sent to live with her reclusive grandfather in the Swiss Alps, quickly grows to love her mountain home—but then is whisked away to the city, where she finds new friends and adventures, but still longs to return to the mountains.
Note: the Project Gutenberg edition (and a second free edition on Amazon labeled 'Gift Edition') is an entirely different translation. The first Kindle edition appears to be the same or very similar to the version I read and loved as a child, and my own personal opinion, after comparing the opening page of both versions, is that it's the better one. Your choice!

Project Gutenberg
When timid Elizabeth Ann's aunt becomes ill, she is sent to live with cousins she has never met on their Vermont farm. Though she dreads it at first, Elizabeth Ann—now called Betsy—gradually comes to love her new home and discovers confidence and adventures she could never have dreamed of.
A free edition is not currently available on Kindle.

Amazon | Project Gutenberg
A lovely old-fashioned piece of children's historical fiction based on the life of Richard the Fearless, who becomes Duke of Normandy as a young boy and must face the challenges of growing up as well as political enemies.

Project Gutenberg
"Oh, what I know about myself isn't really worth telling," said Anne eagerly. "If you'll only let me tell you what I imagine about myself you'll think it ever so much more interesting."
One of the most beloved characters in literature, red-headed orphan Anne Shirley (Anne with an e, that is) transforms the lives of Matthew and Marilla Cuthbert and the little town of Avonlea with her zest for life, her vivid imagination and her knack for getting into comical scrapes. Anne and her adventures are nearly impossible to ever forget.
A free edition is not currently available in the Kindle store, though there are a number of inexpensive ones.

Amazon | Project Gutenberg
Hans and Gretel Brinker dream of entering the great skating race with a pair of silver skates as prize, but with their invalid father, who lost his memory in an accident years ago, unable to work, they are too poor to buy skates of their own to enter. Meanwhile, a group of boys from the neighborhood embark on a skating trip to the capital city of Holland, and eventually their adventures intersect for a happy finale. A lively, funny, heartwarming children's classic you won't forget!

Amazon | Project Gutenberg
One of the best-known and beloved girls' classics, following the trials and triumphs of sisters Meg, Jo, Beth and Amy as they struggle with being poor, befriend lonely neighbors, battle their own besetting faults and eventually fall in love.
The free Kindle edition seems to be cleanly formatted, but on comparing it to a print copy I noticed that a number of the italics and hyphens are missing from the text.

Amazon | Project Gutenberg
Twain's classic, humorous tale of Tom Sawyer, who finds endless amounts of mischief to get into in his small Missouri town—earning scolding after scolding from his Aunt Polly, becoming smitten with new classmate Becky Thatcher, getting mixed up in a murder trial and searching for hidden treasure with ragged Huckleberry Finn.

Amazon | Project Gutenberg
The delightful children's classic about the boy who never grows up, who whisks the three Darling children off to Neverland for adventures among the Lost Boys and battles with the pirates led by villainous Captain Hook. (The Amazon edition is under the title Peter and Wendy, but it is the same book.)
After their father suddenly goes away from home, Bobbie, Peter and Phyllis and their mother move to the country. The three children become fascinated with the railway that runs near their new home, which leads to their making many new friends and eventually solving the mystery of Father's disappearance.
According to customer reviews, the free Kindle version of this book is missing the poems written by the characters in the story, so I've linked to the Project Gutenberg edition only.
One of the most charming "talking animals" books for children, The Wind in the Willows chronicles the adventures of the Mole, the Water Rat, the Badger, the pompous Mr. Toad, and others, by woods, village and stream. (An alternate Project Gutenberg edition contains color illustrations by Paul Bransom.)
The classic children's story of orphaned Mary Lennox, sent to live with her uncle in a mystery-filled mansion on the edge of the Yorkshire moors, where she becomes fascinated with the story of its walled garden that has been locked and the key buried. (An alternate Project Gutenberg edition contains illustrations.)