You might want to crosspost this to r/askfoodhistorians.

They were sucking on plain old limes that had been soaked in a brine solution! Most of the pickled lime/lemon recipes that I've seen from 1700s and 1800s cookbooks feature recipes with vinegar and salt, and no sugar to finish. These recipes seem to be distinct from lime preserve, which featured sugar as a large component. Since both terms are used frequently in cookbooks, home management texts, and so on...it seems pretty likely that the limes Amy was obsessed with were of the sour variety!

In an 1890 edition of the Journal of Materia Medica, they actually suggest making a refreshing drink of lemonade with salt instead of sugar, "...suggestive of pickled limes, a delicacy dear to all schoolgirls." Blech!!!!

The pickled limes were quite inexpensive, at least partially because of the tariffs being very low on imported limes. The fact that they could be purchased so cheaply, often for a penny apiece, made them accessible to schoolchildren. There were actually arguments before the House Ways and Means Committee in 1909 defending pickled limes against a fresh fruit tariff since they were pretty much only eaten by women or children and did not have the value of fresh fruit. Here's a snippet from William Brexnax’s argument:

"After limes are immersed in sea water for twenty four hours it causes such a physical change that they are of no commercial value whatever other than as a pickled lime because they can not be freshened out or used only for eating in their changed condition and in this form they are consumed mostly by women and children of this section of the country who have acquired the taste for them." (Hearings, Volume 20)

There's a cool book called "The Joy of Pickling" by Linda Ziedrich that has a whole section on pickled limes and their brief burst of intense popularity with schoolchildren in New England in the 19th century. According to her, the limes were imported from the West Indies packed in sea-water or brine and shipped to the East Coast in large barrels. New England schoolteachers and doctors greatly disapproved of this trend and even considered it dangerous. "Doctors tended to disapprove of the limes, too; in 1869 a Boston physician wrote that pickled limes were among the 'unnatural and abominable' substances consumed by children with nutritional deficiencies. Parents, however, seemed generally content for children to indulge themselves in the pickled-lime habit” (Ziedrich, 77) 

The Journal of the Royal Horticultural Society, in 1907, has an article about the West Indian lime which specifically mentions, a few times, the increased demand for limes in general in Britain and New York, as well as specifics about the pickled lime process. According to this article, the West Indian green lime export had increased demand following several "fruit shows" and were starting to become better known and be listed as exports to England in the late 1880s.

"There is also fair demand in the New York market for pickled limes. The ripe limes are soaked in sea water for some time, the water being changed from time to time, and finally shipped, in a strong solution of bring, in hogsheads. The first record of 'pickled' limes being exported appears to have been in 1878, fruit to the value of L68 being shipped. In 1906 this product reached the value of L606." (Brooks, 182).

There are lots of references to these super sour treats in various ladies magazines of the period, as well. Here's a little snippet that testifies to the awful pucker-face that apparently went with the eating!

"How can you eat them, Dora!" And Minnie pursed up her rosy little mouth with an expression of decided antipathy. But Dora had purchased a number of the acidulated dainties, and replied, as the two passed out on the paved sidewalk: "I think they are delicious. I always keep a supply on hand, and am going to tease papa to import a cargo of them for my own especial appropriation. George says I'm always eating them; and if I go on, he shall call me "Lady Weazenface;" they pucker up one's mouth so, you know." (Janverin, 312)

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Sources:

Brennax, W. F. WM F BREXNAX IMPORTER BOSTON MASS WISHES A SEPARATE CLASSIFICATION MADE FOR PICKLED LIMES, 20WM F BREXNAX IMPORTER BOSTON MASS WISHES A SEPARATE CLASSIFICATION MADE FOR PICKLED LIMES (1909). Boston, MA.

Brooks, A. J. (1907). The West Indian Lime (Citrus Medica Var. Acida). The Journal of the Royal Horticultural Society, 32, 182–182.

Janverin, M. W. (1859). The Embroidered Handkerchief: Or, How a Piccolo-Maniac was cured. Godey's Lady's Book, 58, 312–321.

Ziedrich, L. (2012). The joy of pickling: 250 flavour-packed recipes for vegetables and more from garden or market. Boston, MA: Harvard Common.