Hi Lollypops,
What type of hardware will your son be using?
Streaming or recording footage can be quite CPU intensive at times, so dependent on the system depends on the quality you get.
The first one is fraps. It is your basic screen capture software which fits in your price range. This can be quite intensive and takes up a LOT of space on your hard drive before you have compressed it.
Open broadcaster software (OBS) is the second option. This is primarily used to stream what you are playing/doing to something such as twitch. You can however set this up to save to your system and then at a later date, upload it to youtube. I will warn you with this however, it can be quite in-depth on setting this up and would need a fair bit of research to go into it.
The last two options are the recording software build into your graphics drivers. Geforce experience and raptr. These are easy to set up and are not nearly as intensive as others but also produces lower quality.
Issues with this i can see is that your posting in the laptop section and i'm betting a fair amount you've got a basic laptop with integrated graphics which won't have geforce or AMD drivers on.
Those are your most popular options. Just be careful with what your son will be uploading, give it a check over and make sure he doesn't accidentally release personal details which were captured on screen/audio.