Gazebos are nice accents in a yard or park but lose a lot of their appeal when the roof starts leaking. The first requirement of a roofing material is that it is durable and waterproof, and the second is that it has an aesthetic appeal that is compatible with the setting and the style of gazebo that it's on.
For portable and temporary gazebos, such as those that many people have set up in their yard for the summer, a canvas roofing material is probably the most appropriate because it's lightweight and easily moved on and off the gazebo. Gazebos with canvas tops that are easily assembled and disassembled can even be taken with you when you go camping or travelling. Treat canvas with a waterproofing compound to ensure that it keeps the rain out of your gazebo and to help to fortify the canvas against mould and mildew.
Wooden and rustic gazebos look best with cedar shingles. Commercially bought cedar shingles are quite expensive, but they last a long time, and they look attractive on a roof. If you have the time and the inclination, you can make your own cedar shingles with an axe, a froe and a shaving horse, which is quite time consuming but much cheaper than commercial shingles. You can buy a large cedar log, and spend some time in the yard turning it into shingles.
Once the dominant form of roofing in the British Isles, thatch is rarely seen today, and, when it is seen, it is usually on the roofs of very expensive homes. Thatched roofs, ironically, have become the purview of the rich when once they were on nearly every cottage in the land with slate reserved for the rich. If you are looking for a truly unusual effect for your gazebo, you can find a thatcher who will do the work for you, or you may even find someone who will teach you how to do it. Don't underestimate the difficulty; effective thatching to create a waterproof roof is an art not easily mastered.
Metal sheet roofing may be more prosaic than thatch, but it is also far more easily available and a lot cheaper, and it's guaranteed waterproof as long as you put it onto the gazebo correctly. Metal sheets can be bought galvanised or painted with a ceramic, rust-resistant paint with the latter option being more expensive but also better looking and longer lasting.