Could it be Stanley and the Magic Lamp, book 2 of the Flat Stanley series?
Stanley Lambchop has found a real magic lamp with a genie inside! The genie tells Stanley he can wish for anything in the world: fame, a magical pet, superpowers -- you name it.
When Stanley's family finds out about the genie, they have some wishes, too. But the more wishes Stanley makes, the stranger his own life becomes. Oh, poor Stanley!
According to Wikipedia, he does have a sibling, a brother named Arthur. One of the reviews mentions an "askit basket":
I loved the idea of an "askit basket." Kind of like a portable, magical librarian.
As does this one:
Text-to-text connections with the Askit Basket in this story and Dan Gutman's book, The Homework Machine.
Also, this Amazon review:
I read a good book called Stanley and the Magic Lamp. Stanley finds a lamp and wishes to get answers for his homework. Then he gets an askit basket and puts his paper in and gets his answers. This book is very entertaining and funny. Read the book to see what happens when Arthur, Stanley's little brother, and Stanley make more wishes from the genie. See what happens when he starts wishing for bigger wishes.
On a side note, the phrase was used as far back as a 1930s game show:
Host: Jim McWilliams (through 10/40), Ed East (from 10/40)
Sponsor: Colgate Toothpaste and Shaving Cream, Palmolive Soap
Type of Show: Game Show
NB: This is my favorite game show of the Golden Age of Radio. The premise was simple. There was a quiz. If the contestant didn't know the answer, the constestant would have to do a stunt which was sent in by the home audience. Some people would do anything for a six month supply of soap, toothpaste, and shaving cream! Ed East was probably best known with Ralph Dumke as Sisters of the Skilletor the Quality Twins.