The South Rim of the Grand Canyon can receive over a foot of snow in a good storm. If you live close by, or happen to be around during this winter event you'll wonder why there isn't a Nordic Center on the Rim. This can be a unique way to enjoy the solitude the winter months bring to this bustling destination, but snow does tend to melt off the South Rim fairly quick after a storm.
The North Rim on the other hand actually gets a snow pack due to its average elevation of over 8,000 feet. The Road to the North closes after the first snow (usually some time in later November) and does not open until May 15th. The Park and all facilities close on October 15th due to the winter weather, so anyone attempting a trip to this area needs experience and preparation. A Winter Rim to Rim is popular with hardcore Canyon hikers, and entails a 30 mile Nordic Trek from Jacob Lake to the North Kaibab trailhead, followed by a 24 mile Rim to Rim hike across the Grand Canyon. There are also limitless forest service roads, as well as a couple trails on the North Kaibab ranger district. These are great areas for cross country skiing and snow camping with numerous places to access Grand Canyon vistas filled with solitude.
The North Rim offers the best snow pack, and therefore the best cross-country skiing, but the South Rim can be fun on the right day. The best places to ski are rim trails and forest service roads that go unused in the winter season. The North Kaibab Ranger District, South Kaibab Ranger district, Grand Canyon Trails Illustrated Topo, and the BLMs Arizona Strip Maps are your best bet to find your way. Giving the National Forest to NPS a call to check on current conditions is also a must do.