As described by Lucie Armstrong in The Ball-Room Guide (London and New York, c1880):
The Calico Ball is a fancy ball at which the dresses are made of calico. Sateen, chintz and velveteen are allowable, and any other material which is made of cotton. The invitation, of course, states the nature of the ball.
It really seems to have been primarily about the fabric rather than any costume theme, though obviously some costumes will work better when made out of cottons than others. She goes on to make some suggestions. For ladies: a dairymaid, a charity girl from St. Giles’, or a Dresden shepherdess. For gentlemen: a Maltese peasant, Albanian costume, Saxon dress, or an Italian peasant.
The anonymous author of Masquerades, tableaux and drills (New York, 1906) added more details: