Since I've been asked twice now:
"weak side" is your bow arm side
"strong side" is your string hand side
:D
If you're unsure of which side you like it on, just make the quiver to look good on either side
The other option, and one I constantly recommend to people who want to make one of these quivers, is to do a mock up.
Take an old pair of jeans and cut the bottom of the leg off at arrow length. Sew the bottom closed (either flat or with an oval added to the bottom) slide in a dowel or arrow and attach a rope for a strap. Presto! A plains-style quiver that took only minutes to make and you can play around with it until you find exactly what
you like! Save the expensive leather for when you have the design down.
Good elk or moose leather in the 6-8oz range is usually $6-9 per sq foot. Garment tanned cow in the 5-7oz range is generally a little cheaper. Something I've been playing with is the 2-3oz garment tanned cow. It's tough, readily available from upholstery shops at only a few dollars a sq foot and should hold up well.