Do I assume that one of each would lead to endless arguments? Also one does not really hack a DS as much as buy a flash cart, though if it is for the DS/DS lite then that is OK as you can get a cheap one that will play pretty much every game (if you do not want to buy a couple of DSTwo flash carts then make sure it is compatible with AKAIO or Wood) and a couple of microSD cards to hold them.
How old are your kids? There are many games that skew younger but the DS I would probably suggest to do better with slightly older ones, though depending upon how much of a fad phase your kids see at school then an old DS/DS lite might not do as well as a shiny 3ds which is more annoying to hack (not to mention in my opinion has a terrible game library).
By fire pads I assume you mean the amazon kindle/fire range which are slightly locked down android tablets, you can twist many of them back to normal android tablets but how easy and how well it works can vary a tiny bit. Even without it they have many games, including a tablet version of minecraft which does well for many (technically there are homebrew versions for the DS as well, and for my money play about as well, though the android stuff has nice easy multiplayer.
As a more general use device (are you anticipating having one in 10 months when people get bored or playing after they have gone to bed?) then the tablets will win without question and for general use I quite like even the stock amazon stuff. The only real hesitation I have is controlling them is not the best (no buttons unless you have a bluetooth controller, and though there is a lot of support for such things it is not on every one of them and thus devs tend to make assumptions).
Anyway without knowing your kids I am not going to quite say that you could not fail either way but it is close, and I say that having played most of the DS library already. If you reckon you could pull off one of each (maybe each having their own flash cart for the DS stuff just to save fights over who overwrote another's save) then that would be the better route from where I sit.