As others have mentioned, a typical smartphone mic is sufficient for most purposes, especially if the intent is more for preservative purposes than high audio fidelity.

If you're recording outside of a studio (ie, in a home, outdoors, etc) make sure the sound in the room is good. Try to pick a space with good acoustics and lots of items to deaden the sound, like couches. If possible, try to hang blankets/fabric over walls to minimize audio echo.

If you're wanting something nicer, some libraries (in the US, at least) will loan or rent a basic digital camera. Depending on the format, you can also buy a wired microphone. (Wireless also works but can be harder to set up). This would probably be the cheapest at-home option to get good audio fidelity, plus you get the added benefit of watching them play.