Without using a cron job, I guess the easiest solution would be to add "a day" to each product. Then select the product of the day with a SQL-Statement using of the current date with now() and LIMIT 1.
You can add a random date to the product as well by using a php script when inserting or updating into the database table. Also you could export the product list to Excel and add the random date to each product as an initial start there.
I did something similar with an advent calendar.
Hope that helps.