Marble cake (cake with light and dark swirls) has unknown origins. It’s first cited in print in an Illinois newspaper of 1859; in 1863, it was mentioned at the Illlinois State Fair, by a woman who also baked silver cake and gold cake. Dr. Chase’s Recipes (1867)  has a “Marbled Cake” recipe that uses white sugar for the light part, brown sugar, molasses and cinnamon for the dark part.
 
Plastic-wrapped marble cake slices are sold in many convenience stores and supermarkets.
 
 
(Oxford English Dictionary)
marble cake n. orig. U.S. a cake made from two mixtures of contrasting colours, swirled together to produce a marbled effect; also in extended use.
1871 MRS. T. J. V. OWEN Illinois Cook Bk. 202 *Marble Cake… White part… Three teacupsful white sugar,..Dark part… Three teacupsful brown sugar, One teacupful molasses, [etc.].
1903 K. D. WIGGIN Rebecca xxvi. 290 She began to stir the marble cake.
1971 M. MCCARTHY Birds of Amer. 74 My husband used to like a marble cake.
1996 J. SHREEVE Neandertal Enigma vii. 184 The sediments of Kebara are thick with these ancient hearths, so dense in places they turn the wall of the excavation into marble cake, with swirls of black and gray ash lacing the brick-red sediments.
 
29 September 1859, Illinois State Chronicle (Decatur, IL), pg. 3, col. 2:
...Marble cake,...
 
Google Books
Transactions of the Illinois State Agricultural Society
Volume V.—1861-‘64.
Pg. 263:
OFFICIAL LIST OF AWARDS
AT THE ILLINOIS STATE FAIR OF 1863.
Pg. 281:
Silver Cake—By Mrs. A. C. Cornman, Decatur, Ills. Commended.
Gold Cake—By same. Commended.
Marble Cake—By same. Commended.
   
Google Books
Tit-Bits:
or, How to prepare a nice dish at a moderate expense
By Mrs. S. G. Knight
Boston, MA: Crosby and Nichols
1864
Pg. 82:
Miss Hull’s Marble Cake.
Three and a half cups of flour, three cups of white sugar, one cup of butter, half a cup of cream or milk, whites of seven eggs, two tea spoonfuls of cream of tartar, one of soda.
Pg. 83:
Five cups of flour, two brown sugar, one cup molasses, one cup butter, two table spoonfuls of cinnamon, one of clove, one of allspice, and one of nutmeg, half a cup of cream or milk,—sour if you have it,—half a tea spoon of soda, yolks seven eggs. Butter your pans, put in a layer of the last or dark, then a table spoonful of light (the first), and alternate with the dark and light throughout.
     
6 January 1867, Dubuque (Iowa) Daily Herald, pg. 3, col. 4:
Fifteen Persons Poisoned by Eating Cake at a Party.
From the Louisville Journal, Jan. 1.
A great deal of excitement was produced at Stephensport, Breckenridge county, Ky., on CHristmas night, by the discovery that fifteen young ladies and gentlemen, members of the principal families of the place, who had been invited to attend a sitting party, were poisoned by eating what is known as marble cake. This cake is made by dissolving or steeping a spoonful or two of cochineal in hot water and mixing it with a portion of the batter, and then arranging that through the rest, so as to give it a variegated or marbleized appearance.
 
Google Books
Dr. Chase’s Recipes
By A. W. Chase, M. D.
Ann Arbor, MI: Published by the author
1867
Pg. 287:
MARBLED CAKE.—Those having any curiosity to gratify upon their own part, or on the part of friends, will be highly pleased with the contrast seen when they take a piece of cake made in two parts, dark and light, as follows:
LIGHT PART.—White sugar 1 1/2 cups; butter 1/2 cup; sweet milk 1/2 cup; soda 1/2 tea-spoon; cream of tartar 1 tea-spoon; whites of 4 eggs; flour 2 1/2 cups; beat and mixed as “Gold Cake.”
DARK PART.—Brown sugar 1 cup; molasses 1/2 cup; butter 1/2 cup; sour milk 1/2 cup; soda 1/2 tea-spoon; cream of tartar 1 teaspoon; flour 2 1/2 cups; yolks of 4 eggs; cloves, allspice, cinnamon, and nutmeg, ground, of each 1/2 table-spoon; beat and mixed as “Gold Cake.”
DIRECTIONS.—When each part is ready, drop a spoon of dark, then a spoon of light, over the bottom of the dish, in which it is to be baked, and so proceed to fill upo the pan, dropping the light upon the dark as you continue with the different layers.
 
20 December 1869, Cincinnati (OH) Daily Gazette, pg. 6:
A “Lady of Indiana” would like to be instructed in the art of making and baking good “muffins;” also, a cheap “marbled” cake.
     
12 January 1870, Cincinnati (OH) Daily Gazette, “Housekeeper’s Department,” pg. 4:
MARBLE CAKE.
KINGSTON, IND.—If a “Lady of Indiana” will follow my recipe, which I give below, she will find it all one could desire in way of “marble” cake.
 
For the white portion, take the whites of 7 eggs, 2 cups sugar, 3 cups flour, 1 cup butter, 1 teaspoonful cream of tartar, and 1/2 teaspoonful soda, 1/2 cup sour cream.
 
Beat the butter and sugar thoroughly together; add the sour cream and the eggs beaten to a stiff froth; pulverize the soda and cream tartar, and add to the last one cup of flour.
 
The dark portion is made of beating together the yolks of 7 eggs, and 1 cup of brown sugar, and 1 of good maple molasses; add 3/4 of a cup of sour milk, one tablespoonful of cinnamon, 1 tablespoonful of nutmeg, 1 cup of butter, 5 cups of flour; to the last add 1 teaspoon cream tartar, an 1/2 teaspoon soda.
Respectfully,
MISS M. J.
 
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The Godey’s Lady’s Book Receipts and Household Hints
By S. Annie Frost
Philadelphia, PA: Evans, Stoddart & Co.
1870
Pg. 353:
MARBLE CAKE—The White Cake.—Whites of seven eggs, one cup of butter, two cups of sugar, half a cup of sweet milk, half a teaspoonful of soda, one of cream of tartar, three cups of flour. Bake two hours in a slow oven.
 
The Dark Cake.—The yolks of seven eggs, one cup of molasses, two cups of brown sugar, half a cup of butter, spice to taste, one cup of sweet milk, one teaspoonful of soda, two of cream of tartar, five cups of flour.
 
This makes two good-sized cakes by putting in (Pg. 354—ed.) first a spoonful of white and then a spoonful of black, and the next layer alternate.