Best is probably the supercard DSTwo family (original DSTwo, DSTwo+ and sometimes going under 4 in 1). Good luck finding one today.
Most of the flash carts that were competitive with the DSi fell by the wayside after a DSi update. While today you can probably work around it (not paid attention to the DSi side of things, though the 3ds one at least allowed you to downgrade its backwards compatibility and thus dodge some of the protections).
Back on flash carts there are various things with R4 in the name (it became a byword for flash cart during the DS so there are hundreds) that will work simply as flash carts, have some cheats, have a soft reset and go from there (no savestates, cheat finding, in game guides and whatnot that the DSTwo and some of its later stage competitors had).
Build quality varies somewhat (though is rarely high), some have timebombs (they check the system date and if above a certain date will error out, such things can be worked around if they support a custom firmware though, personally I prefer to avoid entirely but that can be easier said than done and it is not like new DS games are coming out so do it once and you will probably be OK) and stock levels for.
I don't know what we are suggesting these days for such a thing but there is usually a list or discussion around here somewhere.
Sadly we don't really have a "$25 buy this and have a good time" pick for a DS flash cart, especially not for the DSi and 3ds. We have been hoping one of the big teams comes back and makes such a thing like they have for the GBA but eh.