Support refresh every new day (as opposed to every 24h)

There is a site where namedays are aggregated from. Each day there is info on which saint's name is honored (or a list with names of more than one saint if it so happens).

Now, there is no point to set this feed to be refreshed more often than 24 hours, since the information changes every day and not earlier (each saint is honored on a different day). The problem is that if I set it to refresh every 24 hours after last refresh, I have to make sure that this refresh is done at (around) 00:01 every day, so that when the date changes the nameday also changes along.

Is there a way to add an option in the 'Minimum refresh period' setting so that it only refreshes if the current date is different than the last refresh date? I know that setting it to 24h will do the trick, but it might be that it refreshes... say at 06:30 and all site visitors that come to the website will still see the saint from the previous day.

I hope I've made myself clear on what I ask here. Can anyone think of a workaround here?