As the first of the month falls on a Sunday, I recall my childhood and being asked if my first words that morning had been "Rabbits" or, should the first of the month be a Sunday, "White rabbits". I was told doing so would invite good fortune, although in retrospect it was more likely to be a way of warding off the reverse.
I have since learned few differentiate between the 1st on a Sunday or when it falls on any other day of the week. Most use one or the other at all times. However this has not helped to uncover the origins. The phrase, at least as it is today, cannot have existed prior to the fourteenth century as until then the animal was known as a cony. Indeed even the introduction of 'rabbit' was initially only used for their young, as evidenced by the 1398 translation where John de Trevisa wrote: "Conynges bringeth forthe many rabettes and multiplieth ful swithe (breed like rabbits)". Later we find Edward Topsell writing in 1607 in The Historie of the Foure Footed Beastes: "If two males be out to one female, they fight fiercely; but they will not hurt the rabbets". It was shortly afterwards that the term applied to adults too.
The saying is first seen in print in 1922, where it is said if the first words out of your mouth on the morning of the first "Rabbit, rabbit, rabbit" a present will arrive before the month has ended. However there seems to be several references to the superstition from around 1800, and in several different English-speaking nations.
Perhaps this will remain as mysterious as the origins of the lucky rabbit's foot, although we cannot rule out the two having something in common. It should be pointed out that it is rather odd that the rabbit be considered so lucky when the other lagomorph, the hare, is seemingly associated with more superstition and folklore than any other other British mammal.