Gazebos are open-sided rooms with a covered roof and floor. The roof may be solid to provide complete shade and protection from wind and rain, or it may be latticework to let in dappled sun light. The floor may be made of wood, ground cover, or flagstones and pavers. Remember to water living ground covers. The roof will block all or part of any rainfall. If the flooring is solid wood and elevated off the ground, the only thing you can grow for ground cover under a gazebo would probably be mushrooms.
Flag Stone and Pea Gravel
A quick ground cover is simply leveling the dirt and placing flagstones surrounded by pea gravel as the flooring. Keep the flagstones or other flat stones close together to provide a stable surface for table and chairs. Tiny smooth river rocks, tumbled glass, or gravel may be used as a replacement for the pea gravel.
Pavers are closely butted up against each other. Some even interlock. Leave space between a series of pavers to plant ground cover. Pavers come in a variety of colors, textures, and shapes.
Creeping Ground Cover
Select a ground cover that grows in the shade provided by your gazebo. Plant it between the flagstones. Another idea is to take two different kinds of contrasting ground cover that tolerate traffic and plant a checkerboard square pattern. Alternate squares of ground cover with squares of pavers. Keep the level of the pavers and ground cover at the same height so people don't trip between the two. Creeping Jenny is a good choice if there isn't much foot traffic. Spreading sedum is another choice.
Mosses, Thymes and Dichondra
Scotch moss and thyme are good choices to plant between pavers or stones. They tolerate light foot traffic, which is what most gazebos receive. Dichondra is another ground cover that does well in partial shade and spreads by itself. It has small round leaves, spreads by runners and can be mowed. It will stay short without mowing but turns into a lush thick carpet of green when it is mowed.
If the ground under the gazebo gets dappled shade or partial sun for a few hours a day, perhaps early morning or late afternoon, there are a few grasses that will grow in that environment. Those include bent grass, red fescue, St. Augustine, and Manila. Under shady conditions, grass needs less fertilizer, grows more slowly, and doesn't need to be mowed as much as grass growing in full sun.