If it is mostly the comedy that she likes, then I'd suggest Jerome Jerome's Three Men in a Boat, Lewis Caroll's Alice books, and P.G. Wodehouse's Wooster/Jeeves books. Mark Twain is also funny, particularly A Connecticut Yankee in King Arthur's Court and his lighter short stories. Actual children's books are also often humorous - Ramona, for instance. (Beverly Cleary?) Baum's Oz books have been neglected lately, too.
I'd also second the recommendation of Terry Pratchett in general; the Discworld novels set in Lancre are the best bet, but there are some mild sexual references there, too - Granny Ogg is the opposite of shy. Pratchett's The Amazing Maurice and His Educated Rodents and his Bromeliad and Johnny series are written especially for children. Seasonally apt and certain to delight young children is his 1000-word short "The Computer Who Believed In Santa".
I'm puzzled by the idea that the Hithhiker's Guide books have too many adult situations. Aside from a couple of references to the planet Eroticon VI and its most famous resident (references which would go right over a 5 year old's head) there are no adult situations - certainly far less of an adult nature than you would find in the Bible or Shakespeare, let alone such notorious volumes as Graves' The Greek Myths, Burton's Thousand and One Nights, Andrew Lang's collections of traditional fairy tales, or even Harry Potter, with Dumbledore's brother convicted of practicing inappropriate charms on a goat, its reference(s) to human-giant couples, sexy Veela (who turn into monsters when angered), gambling, drinking (butterbeer and stronger drinks), and drug use (potions, particularly felix fortis).
There are some other humorous SF books, though none as good as Douglas Adams'. Jasper Fforde is the closest I have seen, and I highly recommend his Tuesday Next series. From the cover: "Fforde's heroine, Thursday Next, lives in a world where time and reality are endlessly mutable -- someone has ensured that the Crimean War never ended for example -- a world policed by men like her disgraced father, whose name has been edited out of existence. She herself polices text -- against men like the Moriarty-like Acheron Styx, whose current scam is to hold the minor characters of Dickens' novels to ransom, entering the manuscript and abducting them for execution and extinction one by one. When that caper goes sour, Styx moves on to the nation's most beloved novel -- an oddly truncated version of Jane Eyre -- and kidnaps its heroine." These are more suited for adults due to the literary references, but kids should like them, too. There is an allusion to sex in one of the last two of the five books, but it wouldn't have had any trouble getting released as a film under the Hayes' Code.
Fforde's subversion of cliches bring to mind the website TV Tropes, which is not only great fun, but better education than any amount of reading of stuffy pseudo-intellectual literary criticism. (Though in the latter category Frederick Crewes' The Pooh Perplex deserves an honorable exception.)
Short story collections such as the "Year's Best SF" generally have some funny stories, and generally having been published in magazines, they rarely have much unsuitable. Collections prior to 1970 are almost guaranteed to be clean, and those of the 30s - 50s are especially so. Short stories suit young readers better than novels. Kids generally agree with Sam Goldwyn: "too much plot and not enough story!"
Some other SF humor authors worth mentioning: Keith Laumer's Retief stories (they start to hit their stride around Brass God, and really get going once the Five Eyed Sticky-Fingered Groakies come on the scene as villains.) Going a bit farther afield, and perhaps not up to a 5 year old's rigorous standards for humor: Alfred Bester, Jack Vance (all great, but humor is seen most in Tales of the Dying Earth's Cudgel the Clever - published in 1950, but verify suitability for yourself), Franz Kafka (well, he thought he was writing humor, anyway. Kids usually like horror, too though. ) J.L. Borges' stories are as humorous in a way as they are thought-provoking. The dialogues between Achilles, the Tortoise and the Crab in Hofsteader's Godel, Escher Bach are also amusing, though most of that book will go over young kids' heads. Rudy Rucker and Spider Robinson are often humorous, though generally not suitable for young children. Steven Brust's Jhreg series often has dry wit, but there is quite a bit of murder. Rodger Zelazny also had a dry wit, though it's hard to really classify him under humor. Early Charles Stross works such as Accelerando and Iron Sunrise have a good deal of humor, but there is one scene in the former which is definitely unsuitable, and perhaps neither would be that apt to appeal to a 5 year old.
Comics shouldn't be ignored, either - Girl Genius, Schlock Mercenary, Freefall, and xkcd are my staples among the webcomics. Heavenly Nostrils is a recent web/print comic that shouldn't be missed, especially by girls - a bright 9-year-old befriends a narcissistic unicorn. The author's previous comic Ozzy and Millie is also a gifted kids' classic. Aside from xkcd, these should be read in chronological order since they have long story arcs.
