What I'd probably do is have a database table with all my quotes, then have a cron job that runs every day at midnight and updates a value in my database specifying what quote to display.
So the cron job would run the randomizing script and a quick query to set that value.
Otherwise if you leave it to cookies or some sort of dynamically ran script you're gonna have more places for failure.
Regards,
*edit* Hell, you could even do it all via flat files if you don't have database access.