I would think smaller panniers and maybe a trunk in conjunction with the backpack. Say, ~5 lbs of books in each pannier, and the balance in the backpack (probably the notebooks... as they'll not fit in smaller panniers anyway)... then the lunch in the trunk-- unpack books from panniers into backpack at school and then pop-off the lunch-trunk on separate straps
For panniers, something like the nashbar daytrekkers or something similarly inexpensive (less than $50 and easily mended)... The idea with these is that they stay on the bike--empty when the child is away.
The trunk can be hard or soft-sided... soft-sided trunks will need rigid lunchboxes, hard-sided not so much... however... soft-sided are easier to throw in the wash to clean when the inevitable spills happen.... or it gets smelly...Also, some lunchboxes could be bungie-ed down, so may not need a trunk...
Overall the budget for such an endeavor, for three riders, would be (excluding labor, bikes, lunchboxes, backpacks--all you should already have) at least $25 for a rack, $50 for the daytrekkers, and another $40 for a trunk... that's around $115 per rider... best case.. in reality, I'd expect it to be somewhere closer to $150 per rider... that's about $450 for the three. Keep in mind that you'll need support for the panniers on the racks if you get seatpost style racks as well... to keep the panniers out of the rear wheels. It is also possible to get some 'water proofing' shower cap style covers that will fit the panniers to keep them dry while transiting and while the bike is parked. Fenders should also be considered. Although, some bikes are provided with proper fenders...
If the lunchbox can be retained with the daytrekker's provisions, and the rack+fenders are already present... the budget falls to just the cost of the daytrekkers...