Lawns are man-made, cultivated creations and, as such, can be high maintenance-particularly under adverse conditions. Nevertheless, they do sequester carbon and encourage biodiversity (providing food and habitat for birds and insects); granite and artificial turf clearly do not.
That said, it's tough to make a recommendation for your situation without actually seeing the site. You may want to bring in a local green landscaper to evaluate the site for you. I can offer a few general recommendations for a yard with low light and poor drainage.
From your description, it sounds like the site is not only shaded, but most likely has runoff issues as well. Lawns like well-drained soil, so if you're going to keep the lawn, start by finding out what's happening to the water when it rains. You may need to regrade the property slightly so that the water is running from the lawn (which does not like standing water) to the trees (which need up to 35 gallons of water a week to be healthy). Dealing with the runoff issue will be the best solution for the health of your trees-no matter what ground surface you choose-so explore this first!
If you can't redirect the runoff, consider building a rain garden. (Read this PDF download from Applied Ecological Services to learn more.)
There are a number of shade-tolerant, cool-season grasses available. The best for your situation is probably fine fescue grass (see this chart at Lowe's website for recommendations). Fine fescue grasses are known to be the most shade-tolerant, low-maintenance cool-season grasses for your region. Grass needs time to establish, and it needs to be fed. Keeping your grass at a height of two to three inches when it is hot will inhibit weeds from developing and will help your lawn survive through drier, hotter times. To learn more about organic, natural lawn care, read Paul Tukey's book The Organic Lawn Care Manual.
Worst-case scenario: your yard just may not have enough sunlight to support grass. In this case, you can either go with a low-light ground cover, or perhaps install paving stones surrounded by moss or another ground cover.
If you have to plant less altogether, a limited area of artificial turf might work for you. However, think carefully about what it is your son likes about the yard before you do that.
As a product of an urban environment, I craved interaction with natural things, even plain old dirt, and spent most of my time as a youth in NYC's Central Park (even though the park was very degraded then). We urban dwellers like everything uniform and clean, but we need to remember that this is not nature's way.