It depends how much you're willing to spend on the flashcart.
The two best choices for the DSi systems are the Acekard 2i and the Supercard DSTwo.
The Acekard 2i is cheaper at $15. When you install the AKAIO firmware, most games will work and games play quite nicely. Most importantly, you can enable a child-safe lock on the Acekard 2i, this prevents them from accidentally delete games or anything. An upcoming AKAIO update should make even more games work very well.
The Acekard 2i needs to be manually reflashed to work on an updated DSi. Easy enough if you have a DS Lite already, but if not, be absolutely sure to order "
Premium flashing + updating service" to ensure your Acekard will arrive DSi 1.4.1 ready.
The Supercard DSTwo is pricier at $35, but it can do a lot more. On the current firmware practically everything will work flawlessly (thanks to it's system of bypassing anti-piracy measures), and it has some nice functions on the real-time menu, if they choose to use it.
To get DSTwo working on a new DSi firmware, simply requires placing a file on the microSD card, that easy. The DSTwo is the most future-proof card out there.
Both these flashcarts accept microSDHC memory (up to 32gb) and can have their interfaces skinned to something your kids will liked. If they like Mario, install a Super Mario themed skin! You can find plenty of skins on www.filetrip.net or www.ndsthemes.com
If you're looking for the most compatible flashcart, currently that would be the DStwo, in the future, AKAIO might make up.
If you're looking for somewhere to buy, I totally recommend Shoptemp for any UK citizens, it's free shipping is reliable (took both my orders about 7 days to arrive) and the prices are very fair.