The thickness and the body structure of the sheet of paper never mind in principle. It must not be too thin but still solid enough. If the sheet of paper is too thick or too solid it is not so easy to fold. But if the paper is too thin or too sleazy the figur loses its form afterwards.

Different sorts of paper:

Reprographic paper/normal paper is very good to fold and it is quite cheap, so that you can train each figure for several times before you use another, more expensive sort of paper.

You can accomplish nice effects with a two coloured paper on some figures. Gift wrap paper is a good example for this. One side is coloured, the other side is white and it is quite cheap and good to fold.

Another good option would be special origami paper. Most of them are also two coloured, quadratic fitted and have the ideal measures; from 10x10cm to 20x20cm.
But be careful and try the sort of paper in the shop directly because some of them are slightly rectangular or too thick.

The Silver or gold paper out of cigarette packets is very good for origami miniatures and has a special effect because of different coinage pattern. Also wrappings of pralines are adequate.

Galssine is slightly transparent, tough and very good to fold. There are structured as well as unstructured sorts of glassine which create a beautiful effect to the folding figure.

Just like the silver paper the soft tissue is quite adequate for origami miniatures. It is available in many colours and very thin.

You can also use parchment paper, a premium outline paper, for origami figures. Please note that the square sheet of paper is not excessive smaller than 15x15cm or 20x20cm and not thicker than 65g because it can break easily.

Last but not least napkin paper can be used. The material is good for easier figures like the 'lotus blossom' because of its instability.