We do history pretty much the same way.
One of the problems with the ancient world is that while we know many of the instruments they used, and can listen to the sounds... in the classical ancient world, we have no idea what kind of music was played on them.
Ancient Greek, for example wasn't spoken, but actually SUNG. Not just a tonal language (like chinese) but a SUNG language (my husband has a degree in classics, and it's one of the perpetual gripes. We can't pronounce it correctly EVER, because we don't know how it was sung). No one knows what it sounds like! We can guess, from the way that jewish people sing to god in prayers (very different from hymns) it's probably similar to Judaic prayers, but we have no idea as to what it actually sounded like.
We know the instruments used thanks to archeologists... amongst others the ancient egyptians, greeks, and romans had pipes, lyres, harps, flutes, drums... but we don't know the kind of music played on them!!! Esp not with a sung language like greek.
We DO have some musical notation from egypt and greece, and we know the rhythm (lyrical greek meter), and some scholars have attempted to recreate it. This is actually a major controversy in the Classics world... because many classicists say it's utter nonsense, while others say of COURSE we can recreate it. Both have rather good arguments to back up their positions. Here are a couple links to the side that believes we can recreate it even though we don't know what the language sounded like (that's not up for debate, we just don't know), and are really just guessing at a lot of the notation (like egyptian doesn't have vowels. The country was called KMT in their language, but we call it Kemet, inserting our own vowels. It could actually have been Kamot, or Koomyat, or Kaymeiat, or Kimiti... like the apples and bananas song - I like to oot oot oot ooples oond boonoonoos- but worse.)
http://classics.uc.edu/music/ this page has some links to recreations done by various scholars or groups.
BUT... because we do have an undisputed record of what instruments they played, one thing you might do is to find the modern equivalents. One of the things that's so fun with my own son is the "We don't know..." parts of school. For example, we thought Troy was a myth for ages and ages... until not so long ago... we actually found it!!! People made up stories about what hieroglyphs were for ages and ages, until the Rosetta stone was discovered. We think certain things all the time (like that women didn't fight as uniformed soldiers in the civil war), and then we dig... and find out either something amazing that adds to our knowledge, or that we were totally wrong about something!!! (Like finding Troy, or that women did indeed fight as soldiers in the civil war! I forget the ratio of uniformed women found in mass graves, but the ratio was higher than we allow in our armed services today! Then when scholars dug deeper, they found that some regiments accepted women as a matter of course, and only a few banned women / women had to pretend to be men to 'enlist'. Most women were found with longish hair, modified uniforms that allowed for breasts -instead of hiding them, and had other feminizing aspects, but were in trousered uniforms and died of battle injuries/infection. Pretty cool stuff. But I digress).
Anyhow... with the modern equivilents... you can find dozens of WILDLY differing types of music by culture. Like a stringed plucked instrument similar to a harp or guitar sounds completely and totally different if it's traditional chinese, middle eastern, or european. Our culture reeeeally shapes our music (which I'm betting is part of why you want to do this!), but it also means we're biased looking at other culture's music. Throw a piece of tradition chinese music (written) to someone who hasn't heard chinese music, and it will sound NOTHING like what it sounds like in china. Even when we have the same instruments and musical scores, our brains interpret how it "should" sound differently.
Not sure about your son, but with mine, the "There's still so much more to discover!" really sets his mind on fire, and started him reading critically at a very early age. This is what we know v this is what we think.
Some cool ancient world links
http://www.bbc.co.uk/schools/primaryhistory/
http://www.minimus-etc.co.uk/
http://www.ancientegypt.co.uk/menu.html
http://www.nationalgeographic.com/pyramids/
http://www.ancientgreece.co.uk/
http://www.ancientscripts.com/
http://education.nationalgeographic.com/education/?ar_a=1
BTW... wondering if you're using the actual Illiad, or the super cool kid story version by Rosemary Sutcliff? http://www.amazon.com/Black-Ships-Before-Rosemary-Sutclif...