Hi 'tripper
It depends on the type of leather you will be using.
1. Rawhide - from the little I've seen, it's been twisted before using. (I've never worked with rawhide)
2. Carving Leather - A very good quality leather, can be dyed/stained (uBeaut's water dyes, leather dyes & stains). Comes is a number of 'standard' thicknesses. 2mm, 3 mm, 5 mm etc etc i.e. thin, thick, thick and heavy etc
3. other types of leather that are finished already - eg the leather I use for the coloured notebook covers.
4. Upholstery leather - can be a mix of leather and non-leather used to get the larger sizes for upholstery (again, an area I have extremely limited knowledge and experience)
5. Hides that have the fur/coat left on, eg kangaroo etc - again, I haven't gotten around to using these yet, but imagine they would be like #3.
An ideal type?
If using carving leather, stain it first, seal it, then cut it, wet it and then weave it wet so that it dries into place. You may prefer to stain it afterwards as well.
If staining leather, expect a few imperfections. You cannot seal the leather first like you can timber in regards to staining it.
If you dye/stain carving leather, it is essential you give it a couple of sealer coats as well.
If using another type of leather, go for the best quality you can find/afford.
Try to use the leather from the shoulders and middle of the back, rather than the thinner and finer (weaker) leather from the edges of the hide.
Leather is sold either in hides, half-hides or by weight. Naturally, the bigger the hide or the thicker, the more it will cost. Sometimes you might find a 'seconds' hide where it has a rip or tear in it, or other markings on it. If you can work around or with these imperfections, it's a great way to get a better quality leather for a much cheaper price.
Ideal thickness?
Thickness of the leather itself? definitely depends on the chair and the leather available.
Thickness or width of the stips to be used to weave the seat? depends again on the chair, the leather available and the end look you are after.
If the leather is thin and the strips have a small width, you could be in for a seat with a short-life. I would think starting out with leather that is 3 mm thick as a mimimum would be a good starting point.
A supplier?
Check your local saddlery. otherwise, there are a few places that sell leather via the internet as well. If you are interested, I'll see what I can find (I lost all my bookmarks recently
If you want a few samples, let me know, I've a few scraps that might help you decide what you are after.
cheers
Wendy
Need advice on weaving a leather seat. supplier etc ? 
I've seen it done.A friend's daughter bought some flat pack Shaker chairs back from US. They have some cushion stuff in between the top and bottom, cos that is the way they weave them. Maybe there is a Shaker site with how they do it? 
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