It may originate from as early as the Civil War, when soldiers would jot a list of their items to be laundered (and hopefully returned). In any event, it comes from the late nineteenth to early twentieth century when many sent their laundry out to be cleaned.
Columnist Ed Quillen who worked in a commercial laundry shop as a boy wrote in the Denver Post June 6, 2008, that the term refers to what must be listed when you send your laundry to a service, since 1) there were so many distinctions between different pieces of clothing for men, women, children, and various professions, and 2) the laundry service had to mark every individual piece of clothing per customer, sort into piles by color and water temperature, and ensure that all the pieces were returned to their proper owner.
A laundry list could be very personalized per customer, and might have been frequently re-used for loyal customers. Those with long laundry lists had more clothes and were probably more wealthy than others, and all their belongings would have been more difficult to track.
Through the years the term has come to mean any relatively long, detailed list of items.
It has also come to have a negative connotation. You may find usage such as "She recited a long laundry list of familiar complaints" or "His application had a long laundry list of accomplishments" or "We have a laundry list of tasks" - smacking of dullness and overkill when a focus on fewer high-impact priorities may be more effective.
An exception may be when the term is used as an organizing principle. Some commercial enterprise business system and Web storefront software like DDMS use "laundry list" to mean a categorized list of items that a customer frequently buys, particularly in the office supply, janitorial/sanitation, machine & equipment, or hardware industries. In this case it is similar to a customer pricing contract but more focused on frequency of use, and similar to a Favorites or Wish List but more focused on what has actually been purchased.