A few devices exist specifically for the purpose of dumping these things, such as the NDS Backup Adapter Plus - I don't have any experience with that particular device, but as I understand it, you just put the game into the adapter, and connect the adapter to your PC. This is likely to be the easiest way to do this, but looking around, it seems like it might also be a little more expensive.
Alternatively, if you have a DS Phat or DS Lite with Wifi (DSi or 3DS will not work), you can use a flash cart to do this in a completely legitimate way: You can use a homebrew program to dump and restore saves onto official cartridges. I've personally had good success with Savsender.
You use this program to send the save data to your computer; if everything goes well, your wife can safely delete your saved game, and start a new one. When you want to get back to your own file, you dump your wife's save to a different file (assuming she wants to keep playing her file, otherwise you can skip this step), and restore your own. For a more detailed guide, follow this tutorial.
It's not completely fool-proof: it may occasionally misdetect the save size, in which case you only get a part of your save. Various online databases have the save size available, in case you want to verify the results. In the case of Etrian Odyssey, it uses 64KB for save data (65536 bytes).
There are a few games out there which are known to not work with this program: Pokémon HeartGold, SoulSilver, Black and White are wired differently, and can't be dumped using this process. Additionally, it may not support save sizes greater than 512KB (The Legend of Zelda: Spirit Tracks, WarioWare: DIY and Art Academy are the only non-Japanese titles I'm aware of; Japan had a few more games like this) - I haven't actually tried dumping any of those myself, so I don't know if they're detected correctly. If you want to dump any of those, you are probably best off with the adapter.