I was able to find the unfortunate, racist origins of "step on the crack, break your mother's back" (involving variants of her or your future children being Black), but not the reason that stepping on cracks was considered unlucky in the first place. The previous saying dates to the 19th century, but is the superstition older? I can't imagine people walking cobblestone streets being able to avoid the cracks without playing a never-ending game of hopscotch...
“Step on a crack and your mother will turn black” is not the origin of the phrase. Iona Opie states in The Lore and Language of Schoolchildren (1959) that this phrase was indeed common in some areas of the U.K. in the 1950s but it can be traced back to much earlier in print as early as the late 19th century such as:
“Step on a line, break your mother’s spine Step on a hole, break your mother’s sugar bowl Step on a nail, you’ll put your dad in jail”
At the same time as the derogatory phrase was in use (mainly the 1950s) the widespread saying was that you’d be chased by bears. This deviation came from A.A. Milne in his poem “Lines and Squares.”