Leon,
Bark tanned kangaroo leather is by far the best leather for braiding on the planet. Strands can be cut very thin, under 1/8" if wanted. There have been bullwhips made with 36 or more strands in the overlay! See David Morgan's book
Whips and Whipmaking. Kangaroo leather is a stiff tuff leather, very thin, and so strong that even fine strands have incredible tensil strength. I've made a sling with a kangaroo hide pouch and bootstring arms. I made it back before I had a good supply of leather, years ago, so I didn't try the plaited (braided) thongs at that point. The braided thongs should make you a first class sling. When done right they won't twist, thus eliminating all tangling during use. I remember when I made my first braided sling out of yarn. Because it didn't twist in use, my releases became so smooth. I had been used to the boot strings and this was so much nicer. The braided leather will be the same. It's durability should last for years too.
Deer skin can tend to be spongy and soft and very difficult to cut smoothly or braid tightly. I've heard that indian tan buckskin, though similar in some ways, is very very durable.
If your local leather supplier doesn't have bark tanned kangaroo leather, you can get it from
www.siegelofca.comor
www.davidmorgan.com Siegel has more color choices. Drum stuffed is better than greasy but both will work.
As I see it, you can use several different techniques to make such a sling but if you want the thongs to come directly off the pouch without attaching them later, then cut out something like this from the perimeter of a trimmed and stretched hide.
=============<>===============
The === signs represent 4 strands coming off the pouch <> on either side. Just cut it out like that directly from the hide. Make your strands on each side 1.6 times the finished length you'll want them to be. Stretch the strands before you start. Do four strand round braids out of them and pull them tight as you go using lard to make them slide in place and tighten up. It's best to attach the pouch to a clamp or hook on the wall, so you won't have trouble pulling as tight as you can on every pass. Then finish off with knots on either end. You should be able to make a loop at the one end by some means. Finally, finish it by putting the round braids down on a piece of plywood and rolling it under a 2by4 until smooth. Dry it out and you're ready for business! Morgan's book has instructions on the 4 strand round braid, though it shouldn't be hard to find instructions elseware in any braiding book. It is possible to do the braids with more than 4 strands but I wouldn't recommend it for sling application. I would recommend you cut each strand a little more than 1/4" wide. Maybe 5/16" each
Here is a picture of my favorite sling, with my favorite rock

I made it out of some good yarn and the braids are done with 8 strands over a 4 strand core. They are quite stiff and really nice. The pouch is made doing a knotting technique used for friendship bands and the only draw back is that the edges of the pouch where it transitions into the thongs, is wearing out from the friction of releasing the rocks. I put some glue on the inside as a surface to protect it and it seems to be working. Chris generously put my sling up so I could use it here.

And here it is again, with my baseball, practice sling.

Soon I hope to do some more experimentation. I should try a leather sling myself. I would except that I've been handling leather for 10 years now, and I am more excited about textiles at the moment. Good luck on your sling making. Post a picture for us!
Ben