There are a ton of different answers to this question depending on your budget and your nephew's intended results.
On the low side of things would be an all-in-one package such as the Zoom Q2N, which is a video recorder with integrated stereo microphones. Many working musicians use them for recording rehearsals or YouTube clips. They run around $160.
The upper end of things has nearly no limit, but would include:
1) A full set of drum microphones. The budget sets run about $100, but the sound is pretty much what you would expect from a budget set. Individually buying top-of-the-line microphones for each drum can run into the thousands of dollars.
2) A quality audio mixer with USB out capabilities (or a dedicated audio interface) that has enough channels (inputs) to handle all the drum microphones. This will run anywhere from $200 to $700.
3) A computer with a Digital Audio Workspace (DAW) and video editing software to (a) mix the audio signal and (b) match it to the video shot by --
4) A digital video camera (price range as you'd expect). Look for one that can connect wirelessly or by cables directly to your computer.
There are a lot of threads concerning this common question on this forum, so feel free to use the search function as well. Again, welcome!