According to egullet.com, they are named Bread and Butter, because during the depression they were as common on the dinner table as bread and butter.
That didn't seem right to me, since I remember my grandma saying she had them as a child, and that would have been before the 30's. So I kept searching.
Some sources said that during the depression people made sandwiches from them with bread and butter, (Yuck!) but again, I doubted that was the beginning of them.
The Journal of Home Economics in July of 1929 says that they were eaten with bread and butter instead of being merely a condiment. So, they were around before the Great Depression.
Then I found this on CooksInfo.com :"TWIN CITY PEOPLE in Other Cities. Mr. and Mrs. Omar Fanning were here from Streator Thursday, calling on relatives. Mr Fanning will go east today to close a deal with a large corporation for the sale of his pickle business in Streator. Among all the former Sterling people who have succeeded unusually well in other places, Mr. Fanning's success has been unique. He engaged in the business of raising cucumbers at Streator a number of years ago, and gradually developed a pickle business in order to use the small cucumbers that had been a waste product of his greenhouses. In time the pickle business became the main business and his products have met with immense sales all over the United States. A business regarded as in the millionaire class thus grew out of the thrift that prompted Mr. Fanning to make use of the waste product, and it is said that an old receipt for making pickles that had been in his family for two or three generations had quite a lot to do with the popularity of the "Bread and Butter" pickles which he originated and put on the market." -- Sterling Daily Gazette. Sterling, Illinois. 11 June 1926. Page 2 The Fannings went on to create a small pickle fortune.
In case you have ever wondered where they got their name, GFA Brands, which owns Mrs Fannings, has supplied the background. In the early part of the [Ed: 20th] century Cora and Omar Fanning of Streator, IL found themselves short on cash. What they had going for them, however, was a reliable crop of cucumbers and Mrs. Fanning's great recipe for sweet & sour pickle chips. Mrs. Fanning worked out an agreement with a local grocer, who gave her groceries -- including bread 'n butter -- in exchange for the pickles. The name stuck, and has been used by many companies ... Although Mrs. Fanning's pickles began in the Midwest, they are not widely distributed there, but are more readily found in the Eastern, Southern and Western states." -- Feingold News. Feingold Association of the United States. Alexandria, Virginia. October 1996. Page 5.
All because of Bread and Butter Pickles.