The term “elimination diet” was popularized by Dr. Albert Holmes Rowe (1889-1970) in his books Food Allergy (1931) and Elimination Diets and the Patient’s Allergies: A Handbook of Allergy (1941).  Elimination diets remove a food for a period of time to identify the foods that give an individual adverse effects. “‘Elimination Diet’ in the Treatment of Food Allergy” was a chapter in Rowe’s 1931 book.
 
Rowe is said to have first proposed “elimination diets” in 1926. “The Hauser Elimination Diet” was advertised in 1928 by Dr. Benjamin Hauser, who later became known as the food guru to Hollywood stars.
 
“Exclusion diet”—a diet that excludes certain foods that give adverse effects—is often used instead of “elimination diet.” The term “exclusion diet” has been cited in print since at least 1899; “Allergy exclusion diet” was cited in print in 1973 and the term has become popular since the 1970s.
 
   
Wikipedia: Elimination diet
An elimination diet is a method of identifying foods that an individual cannot consume without adverse effects. Adverse effects may be due to food allergy, food intolerance, other physiological mechanisms (such as metabolic or toxins), or a combination of these. Elimination diets typically involve entirely removing a suspected food from the diet for a period of time from two weeks to two months, and waiting to determine whether symptoms resolve during that time period. In rare cases, a health professional may wish to use an oligoantigenic diet to relieve a patient of symptoms they are experiencing.
 
Common reasons for undertaking an elimination diet include suspected food allergies and suspected food intolerances. An elimination diet might remove one or more common foods, such as eggs or milk, or it might remove one or more minor or non-nutritive substances, such as artificial food colorings.
 
An elimination diet relies on trial and error to identify specific allergies and intolerances. Typically, if symptoms resolve after the removal of a food from the diet, then the food is reintroduced to see whether the symptoms reappear. This challenge-dechallenge-rechallenge approach is particularly useful in cases with intermittent or vague symptoms.
 
The terms exclusion diet and elimination diet are often used interchangeably in the literature, and there is no standardised terminology.
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History
The concept of the elimination diet was first proposed by Dr. Albert Rowe in 1926 and expounded upon in his book, Elimination Diets and the Patient’s Allergies, published in 1941.
     
Google Books
January 1899, The Medical World, pg. 12, col. 2:
I have watcht patients that were held to the strictest exclusion diet, drank skimmed milk by the barrel, mineral waters by the hogshead and swallowed oceans of drugs; and yet they steadily declined and yielded to the natural outcome of the disease.
 
27 June 1928, Tampa (FL) Morning Tribune, pg. 18, col. 5 ad:
The Hauser Elimination Diet
Those of you who heard Benjamin Hauser’s enlightening lectures last winter and know the famous Battle Creek Sanitarium’s dietary system will be interested to know that the identical service is available to you right here in Tampa.
(Florida Health Products Co.—ed.)
 
Google Books
Food Allergy:
Its manifestations, diagnosis and treatment, with a general discussion of bronchial asthma
By Albert H. Rowe
Philadelphia, PA: Lea & Febiger
1931
Pg. 60:
“Elimination Diet” in the Treatment of Food Allergy
 
Google News Archive
4 March 1931, Spokane (WA) Daily Chronicle, “Food Sensitiveness Tests Given” by Logan Clendening, M. D., pg. 4, col. 2:
Another good way, recently proposed by Dr. Rowe, is with elimination diets.
 
How It’s Done.
These elimination diets consist of foods that are not frequently the cause of allergic diseases. The patient stays exclusively on one after another for several weeks. If none of the looked-for symptoms—skin eruption or asthma or nasal discharge—occur, it may be assumed that these were originally caused by some article of diet not on the list. The patient can then avoid all other foods except these on the elimination diets, which are sufficient to keep him in good nutrition.
   
OCLC WorldCat record
Skin tests in 4589 cases of allergic disease with a criticism concerning elimination diets
Author: George Plness; Hyman Miller (Calif USA Los Angeles) Affiliation: Los Angeles, Calif. USA
Edition/Format:   Article : English
Publication: Journal of Allergy, v4 n1 (1932): 18-23
Database: ScienceDirect
     
OCLC WorldCat record
Elimination diets and the patient’s allergies; a handbook of allergy,
Author: Albert H Rowe
Publisher: Philadelphia, Lea & Febiger, 1941.
Edition/Format:   Book : English
 
Google Books
Allergy Management in Clinical Practice
By Louis Tuft
St. Louis, MO: Mosby
1973
Pg. 147:
Allergy exclusion diet
 
8 April 1976, Morning Advocate (Baton Rouge, LA), “IFT Studies Diet, Hyperactivity,” pg. 10-G, col. 1:
Recently, recommendations have been made that hyperactive children should be treated by excluding from their diet all foods containing synthetic colors and flavors, as well as many fruits, berries, and vegetables, according to the IFT (Institute of Food Technologists—ed.) Communicator. The summary addresses this aspect by stating that “The exact additive and nutrient content of the exclusion diet is difficult to assess, since it manipulates a myriad of factors.”
 
OCLC WorldCat record
An Australian exclusion diet.
Author: AR Gibson; RL Clancy
Edition/Format:   Article : English
Publication: The Medical journal of Australia, 1978 Mar 11; 1(5): 290-2
Database: From MEDLINE®/PubMed®, a database of the U.S. National Library of Medicine.
Summary:
Exclusion diets may have a practical place in determining the precipitating dietary factors in certain clinical conditions. We present an exclusion diet which is based on the exclusion of food commonly known to cause food allergies, and the exclusion of food which contains salicylates, benzoates, tartrazine, yeast, and penicillin. This provided a basis for challenge with these additives and natural chemicals. Preliminary information in urticaria suggests a use for this diet in some allergic conditions.
   
The Huffington Post
Gwyneth Paltrow’s ‘Elimination Diet’ Is Hardly The Craziest Diet To Come Out Of Hollywood
Posted: 03/19/2013 8:50 am EDT |  Updated: 03/19/2013 5:02 pm EDT
Gwyneth Paltrow is probably used to getting scrutinized for her unconventional diets, but when excerpts of her new book “It’s All Good” started circling, she was suddenly blamed for being a bad mother, too. But is all the heat really warranted, or were her quotes taken out of context?