I’ve discussed this with colleagues over lunch several times. The idea wouldn’t exactly be random, we’ve discussed a hash of the current date modulo some number

I’ve discussed this with colleagues over lunch several times. The idea wouldn’t exactly be random, we’ve discussed a hash of the current date modulo some number of choices. So for any frequent choices, come up with n options and then take the hash of the date or weekday, or just pick a random number. Then do modulo n to determine the choice for the day. The nice thing about a hash is that the result feels random but could exhibit a recognisable pattern over time.