I first encountered this story in a play by Aaron Shepard, which you can read online: The Gifts of Friday Eve. The version of the story that he tells comes from Iran. A distinctive feature of this story is the reunion of a father with his long-lost son. This version does not, however, feature the incident of the magical steps and the mysterious jewels that Mushkil Gusha bestows upon the poor man.
You can listen to a version of the story told by Idries Shah in this podcast: The Story of Mushkil Gusha | The Idries Shah Podcast. This version features the mysterious steps and jewels, but it does not have the reunion of father-and-son.
You can also see a beautiful PDF of the story here: The Tale of Mushkil Gusha. It appears in his book The Caravan of Dreams, which you can read online at the Idries Shah Foundation website.
Idries Shah's son, Tahir Shah, weaves a version of the story into his book In Arabian Nights: In Search of Morocco Through Its Stories and Storytellers (CDL at Internet Archive). The version of the story here is like the version in Idries Shah's book, but what is really fascinating is that the storyteller, Murad, learned the story from a talking eel: more about that.
A version of the story from Pakistan appears in Tales of South Asia by Beulah Candappa: The Invisible Friend of Man (CDL at Internet Archive). This is a shorter version of the story which ends when the father and daughter realize the value of the "pebbles" and become wealthy. There is no princess or stolen necklace.
You can also find a version in The Illustrated Book of Fairy Tales by Neil Philip: Mushkil Gusha (CDL at Internet Archive). The mother is now included too, and she is the one who discovers the true value of the "pebbles" when she goes to sell onein the market. This version features a man at the end whose son is dying, but after he hears the story of Mushkil Gusha, his son recovers.
Phillip's source was this public domain version in David and Emiliy Lorimer's Persian Tales, published in 1919: Mushkil Gusha. Because this is a public domain version, it will always be available somewhere online! It features a curse as well as a blessing: when a proud man on horseback will not eat and talk with the woodcutter in prison, the father curses the man and his horse to break their bones, and it happens. Then the father of the dying son arrives, and the son recovers while he eats and talks with the imprisoned woodcutter.
Here is a note they include with the story:
Poor and pious women in Persia have the custom of telling the story of Mushkil Gusha on the Eve of a Friday, and in this wise: they fast all Thursday (viz. from sunset on our Wednesday till sunset on our Thursday), and at noon they find a child who has never heard the tale and tell the story to him (or her). If no child is available they put a mirror on the ground and tell the story to the face in the looking-glass. Then when the time of sunset prayer has come they break their fast with pease and raisins or with dates. Whatever is left over from the "break-fast" they distribute as alms in the name of Mushkil Gusha. No one to whom such alms is offered should refuse to accept them, but he may hand them on to some one else. The story as it follows above was taken down verbatim from the lips of one of these poor and pious women by my informant. It is believed that whoever follows these instructions will find all his desires fulfilled. It is open to the reader to put this faith to the test.