Well building you own would be the best way to come close to your desired final cost (and still end up with a quality AR) but $600 will be tough-to-impossible figure to meet, especially using all new components and even tougher considering your fairly short time frame. This is what I would do...
Stripped lower, which ever one floats your boat and is priced around $100.
LPK for ~60 and a decent stock for between $30-$75 depending on whether you go new or used, fixed or collapsible. Building your own AR lower is a simple and beautiful thing. It will be a great project that you and your son can do together and will help him (and maybe even you) see exactly how everything works.
On the high end, you will have about $235 into it at that point leaving you $365. This is where it gets tough. The easiest and probably cheapest way to go would be to get a used upper off of the EE but even then, getting a complete upper for $365 that isn't a POS or shot out will not be easy to find. Most decent used uppers in the EE go for between $300-$350 but that usually does not include the Bolt\Carrier (~$100-$125 new) or the charging handle (~$20-$25 new). Also keep in mind that if you get an A3 upper, then you will still need to figure out something for sights\optics (assuming you don't already having something lying around he can use?).
This is where having the tools to assemble your own upper comes in handy, you can build a new upper for close to what most used uppers sell for. BM barrels sell for ~$200, CMT A3 upper for ~$90, gas tube ~$8, hand guards ~$30
I don't know how much play room you have in the budget but if you can swing it and can't\don't want to build your own upper, you are not likely to find a much better deal then these factory built uppers...
CMT/STAG ARMS - 2H Complete Upper