I like this idea. The school I work at have two days entitled "Muffins with Moms" and "Donuts with Dads". On that day the child eats a muffin/donut with dad, mom, grandparent, aunt/uncle... whoever can (and willing) to come with the child. As they are eating, they can together read their new book together. It is a great idea...
Reading Month in Michigan is in March, so our PTO gives a book to every child, pre-k through 5th, in the two schools we support. We used to order the $1 each books through Scholastic also, but found that if two or three of us look throughout the year, we can find next to new books at rummage sales and thrift stores for way cheaper. I've bought over 500 books for .25 to .50 each this summer. We lay out age appropriate books in the library during their regular library class the first week of March. We just call it The Great Book Give Away.
Boy oh boy, there is almost nothing finer than seeing 25 kids' faces light up when the librarian tells them that they noy only get to pick out one of the book and take it home BUT they get to keep it forever and ever!
Maybe you could call your program...."Just Because Books". Was the intention to pass them out on the children's birthdays throughout the school year?
We are doing this as well, twice this year. We're going to the Scolastic Warehouse Sale and buy books for $1 each and give one to each child for Christmas/Holiday Season, and at the end of the year. I know we came up with some cute name, but I can't think of it right now.
Are you wanting to distribute the books so it's not grouped like a class or grade at a time, but more random?
If so, what about just giving the program some snazzy generic name, maybe associated with an internal library or literacy program. Then distribute based on some other factor - like first name starting with an "A" gets a book in week 1, "B" = week 2, etc.