* the player (he's only just started learning)
* the room (most computers live in a small study with dodgy acoustics)
* the house and street (is everyone going to be perfectly quiet while he records?)
* the guitar (he's only just started, so I suspect it'll be fairly budget, and may not be perfectly tuned either)
* the mic
Until your son is playing seriously and until you're recording somewhere other than the average room in the average house, really don't bother getting anything else. What you've got will be an accurate recording of his playing at the moment. It won't sound like John Williams in a studio, bcos he won't sound like John Williams and your house won't sound like a studio!
When you're recording him, do try putting the mic in different places around the front of the guitar and see where sounds best. Around the neck/body join is traditionally a good place for mics, but play with it and see what works. A mic stand will help here - something like this stand would be fine for not much money.