Green materials are usually natural, made from renewable resources, or use low-energy manufacturing techniques. Some people consider materials sourced and processed locally to be more green than those that are not. Some materials combine these properties, but not all of them do. For example, bamboo cloth is made from a natural and renewable resource (bamboo grows very quickly), but it requires a lot of nasty processing to make it into suitable cloth.
For wood, this usually means avoiding exotic woods that are from endangered species, or have to be shipped from far away. There is plenty of nice renewable hardwoods available from the US. A green home can use salvaged wood (e.g., from old buildings) that has some interesting character to it. Salvaged woods do not need to be harvested, and salvaging saves them from the dump. However, some sources of salvaged wood may have toxins, so it should be obtained from a reliable source.
Marble or quartz is not very renewable, but many manufactured stones are made from recycled materials. Paperstone is made of recycled paper bound together with resins. NovuStone creates counter tops from recycled glass; the glass is sourced and processed in the Pacific Northwest.
A green home would avoid most synthetic carpets. These carpets are made from petroleum producs and use nasty treatment chemicals to make them flame retardant. Natural wool carpets provide a green alternative, and they are naturally flame retardant.
Green homes avoid using paints with lots of VOCs and use low-VOC or zero-VOC paints made from plant oils or natural minerals instead.
These materials often provide benefits for the home owner as well as the environment. Low-VOC and zero-VOC paints improve air quality — the air is cleaner from the beginning, and you don’t have to worry about any possible health effects (VOC concentrations can be up to 10 times higher indoors than out). Wool carpets also reduce the amount of VOCs in the air. They can also be more durable, more stain resistant, and are less prone to collecting dust mites. Plus, they often feel a lot nicer. Of course, this comes with a cost. Wool carpets are significantly more expensive than their synthetic counterparts.
Synthetic and manufactured stones often come in a wider variety of styles an colors than natural stone. Some can be made to custom-order. Some types of manufactured stone counter top can be created as a single slab (which makes cleaning easier).