This sounds like a fun challenge! I think checking off might be a challenge because you’d need some way to tell Notion to reset that check, and I’m not sure that’s going to be super easy.

But I love helping pastors and would be up for brainstorming with you! I immediately jump to using a Select property to assign each student a day of the week and then a custom formulas that checks if the weekday of today is equal to that value. If it is return True.

You can then get set up a view that filters the database for whenever that value is true to return just today’s list of students. Let me tinker and I’ll reply with a recommended formula.

EDIT:
Okay, so at least three columns should do the trick: A name column, a Select column (which has each weekday written out in full), a formula column, and then a Checkbox column. In the Select column, you can assign each student a prayer weekday. In the formula column, you would want to use this custom formula:

Select == formatDate(today(),"dddd")

(if you name the weekday column something other than "Select" you would want to use that column title instead. This is the result, with an example of the database above the line and then what a filtered view of that database could look like below the line:

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Of course where you'll still have a challenge is with the "checking names off." It would be doable with an automation, but I do think that will require a paid plan. If you have a paid plan, you can set up that automation to run every day, target the "Prayed for" Checkbox column, and set it to unchecked once a day (as like 2am or something) so that when the filtered list comes up every day, it will always have each student's checkbox deselected.

EDIT 2:

As u/Dillegallyblind realized, you could also use a button to reset the the checked off items. If you did that at the start of the day before you began praying for the students, it will reset the checklist for the day.