Dear AbercrombieBaseball,
You should not worry. First of all, here is how my DS battery has been working: I play games until the power light turns red, which means I have maybe half an hour left of non-Internet gaming. I normally plug in my AC adapter into my DS and into the wall when the light turns red so I am not interrupted later by my DS just shutting off.
After my cord is plugged into my DS and the wall, I choose to either resume playing (the power light should turn green after a few seconds, and the orange charging light should come on) or turn off my DS and let it charge (only the orange charging light should be illuminated). NOTE: playing your DS while plugged into the wall is, as far as I know, perfectly safe for your DS, and your DS's battery will actually charge a little, although the battery will charge MUCH faster if your DS is off.
That is how I tend to charge my DS. Feel free to use the same methods, if you're unsure of the way you're currently doing it.
Now, as for your situation: I honestly don't know how long it took my DS to fully charge the first time I plugged it in; it was very long ago. Nowadays, it takes around an hour for my battery to fully charge. You will know if the battery is done charging when the orange charging light on the DS is no longer illuminated. At that point you SHOULD unplug your DS. I sometimes let mine charge overnight (I heard that's not a good thing to do, because it might shorten the lifespan of the battery in the long run) and unplug it the next day. Anyway, I'm not a technical whiz so I don't know if my statements are 100% true, but this is what I know: Your DS's battery will not explode or be damaged or whatever if you don't let it charge all the way. The only things I can think of that will affect the quality of the battery are: Overcharging (may cause shortened battery life after long usage), physical damage (dropping the DS, water damage, etc) and uninstalling/reinstalling the battery improperly (I don't recommend you unscrew ANYTHING on your DS, so no worries on that one). Bottom line: Your cousin shouldn't worry about damaging the battery, just make sure to keep an eye on that power light and plug/unplug when necessary for charging necessities.
I apologize if I made mistakes in my statements and if someone sees some, feel free to make corrections. I also apologize if you already know some of the stuff I said, but I figured it would be helpful to include some proper instructions on charging, although it's probably covered in the manual, now that I think about it.