Hi there,
Assuming data is being pulled from a MySQL database, and depending on how the code for the existing widget is setup, this should be fairly straightforward to figure out.
Essentially, make a new table which stores a "model of the day", with 2 fields (model's ID, and the current date).
When the page loads, check the most recent record in the table. If the date on that record is today, show the model matching the stored ID. If the date on that record is older than today, generate a new random model ID, write the record to the table with today's date, and then show that model's ID.
Doing it this way should prevent too much strain on the database. It will only generate a new one the first time the page is loaded that day.
You could have the "model of the day" permanently stored in the database, so that you have a running history, and perhaps create a "Past featured models" page at some point. Or you could just have it save the current day's model, and ignore the rest.
Keeping a history would also allow the option of having a setting like "Don't feature a model more than once in X number of days". Then the first time it's loaded each day and needs to generate a new random model ID, it'll look to see if that model's been shown in the past X days, and if they have, generate a different model ID.
Whether or not I win the bid, I wish you well. I'm a photographer myself, and I know how tough it can be establishing a site like this.
Regards,
John Aldred