Speculation tiem!!
Keeping in mind that at this stage, any flashcart running on a DSi is purely experimental, let's think this thing through to its logical conclusion...
Flashcarts currently available to the public are not compatible with the DSi. This is common knowledge to anyone with the slightest interest in the flashcart/homebrew scene.
The hardware in the DSi is quite different to the older DS models. Also, there's those new "security measures" Nintendo have added. Obviously, it's going to take more than a flashcart software update to achieve compatibility with the new hardware.
So yes, I think new improved flashcart versions will be needed to boot homebrew and/or backups on a DSi.
Perhaps the new models will be compatible with DS Phats and Lites - that would allow flashcart manufacturers to keep production costs down. But cartridge space constraints could make that difficult.
But (and this is the real question at this point in time), if anyone succeeds in getting the DSi's SD card to boot/run code, the flashcart discussion will be moot.
Will I be buying a DSi and a new Cyclo DS to go with it? Only if my current Lite implodes - and due to some unforseen apocalyptic circumstances, I can't buy another Lite to replace it in (at least) the next 12 months...