Reading your question a few possibilities come to mind, but one in particular stands out from the rest?stonedust.
Chances are that if you live in Brooklyn, your backyard is shady. There really is no ground cover that can tolerate foot traffic like grass can, especially in the shade. The closest thing would be moss, but you would have to tread lightly for moss to thrive.
Stonedust is a clean mixture of quarter-inch and smaller stone fines that are created during the processing of bluestone. The mixture of small, angular stones and fines compacts easily into a firm, stable surface that is porous, allowing for groundwater recharge and reducing volume in the public storm-drainage system.
Stonedust is easy to install. A typical cross section could consist of eight inches of compacted subbase gravel (to provide good drainage), topped with two to three inches of stonedust. If your children dig into the stonedust, just level out the surface with a rake?or have them do the repairs! Include native plantings around the edges so that you have a more appealing garden. The plants will also help absorb storm water and provide food and cover for wildlife.
Stonedust is relatively inexpensive, locally produced, and readily available at your local stone yard. It's easy to walk on, it provides a perfect surface for outdoor furniture, and its soft, dark-gray color blends well with the landscape of the Northeast.
Another option that will also provide you with a porous ground surface would be to place a local stone (such as bluestone or fieldstone) on top of the same combination of subbase material and stonedust that I described above. Keep the joints between the stones about four to six inches wide and plant these spaces with native moss or stonecrop (Sedum ternatum). This solution has all the advantages of stonedust alone, but with an alternative look.
You might want to include a sandbox in one corner of the garden to keep your children from excavating the stonedust. Make a raised container (about nine to fifteen inches high) and it can double as a planter when the children outgrow the sandbox.