Random Name Generator

I would probably approach this with a "shuffle and sort" algorithm rather than use RANDBETWEEN() in an INDEX() function.

1) From your list of 21 names, create a list of 105 entries, where each name appears 5 times.
2) Assign random unique numbers to each entry in this new list (I tend to use RAND() for this. It's not guaranteed not to repeat, but, IME, it only very rarely repeats). Remember that RAND() is volatile and will recalculate with each calculate event. You may consider something to control when this list of random numbers refreshes.
3) Sort the list by the random numbers.
4) Fill the 7x15 grid by reading names from the top to the bottom.

I've left it to you to fill in the programming details at this point. Do you need help with the programming details for each step?