It was a common belief among RAF bomber aircrew during World War II that saying “white rabbits” upon waking would protect oneself.
There is also this passage from the 1922 novel Solomon in all his Glory by Robert Lynd, which suggests a more materialistic reason: “Why,” the man in the brown hat laughed at him, “I thought everybody knew ‘Rabbit, rabbit, rabbit.’
“If you say ‘Rabbit, rabbit, rabbit’—three times, just like that—first thing in the morning on the first of the month, even before you say your prayers, you’ll get a present before the end of the month.”
Another theory is linked to that other traditional playground saying ‘a pinch and a punch for the first of the month’.
Once someone had inflicted their pinch and punch on an unknowing victim, saying ‘white rabbits, no return’ meant the victim wasn’t allowed to pinch and punch their attacker back. Therefore, saying ‘white rabbits’ as soon as you awake on the first is supposed to protect you from those pesky pinchers and punchers all day long.