Upgrades in the computer world typically add features, and additional features almost always take additional memory. That applies to iPhone iOS upgrades.
8GB iPhones are not manufactured any longer, so, yes, memory must be extremely tight. Further, iPhones (and iPads) need an absolute minimum of 10% of their space free so iOS can do its work. In your son's case, that means at least 800MB must be free or the iPhone will not be able to function.
Depending on which version of iOS is on the iPhone, try Settings > General > Usage to see what is using much of the space on the iPhone, and clean off some of the heavy hitters.