The strapping shouldn't matter that much since it shouldn't be seen at all. The type and thickness depend on what you personally put it through in terms of required durability. The straps should be leatheresque, meaning so long as it looks like leather (and not hi gloss) then it only comes down to, again, durability and what hell you are going to be putting it through.
When I build my armor, I fashioned a harness out of 1.25" strapping in order to keep all the sections of the armor fitting nice and neat against me. It has the straps that go over the shoulders (though under the shirt), around the chest area, around the waist area, and around the hip area. This keeps the various sections from flaring out funky.
On the straps, I still use Rob's from my old MWA kit before I built mine since it's a good piece and I haven't had time to do my own, but if you are redoing straps, I would suggest using something meant to hold up to wear and tear like the faux leather upholstery stuff that thick and sturdy, as well as coming in several textures.