3/4" plywood is my suggestion.
Plain plywood here is southern yellow pine. Structurally strong stuff, but kind of soft and rough on the surface. What I've seen happen is chips and shavings get stuck to its rough surface and get pressed into it. With a soft workpiece such as white pine (something I've been using a lot of in a few recent projects), that leaves an embossed dent in your workpiece where the chip is sticking out. So yeah, the pine jaws themselves will not damage your finely finished workpiece, but can lead to damage.
I recently found myself wanting to use my benchtop Workmate, which has aluminum jaws. I added leather jaws to it by using screw in snap bases screwed into the plastic side of the jaws, and pressed snaps into a leather belt blank. That is perfectly safe even on white pine, and has fantastic grip.
If you have full grain cowhide available, that would be a great jaw covering, though it would probably get beat up over time more quickly than other options here, and is more expensive too.
Vise soft jaws are made to be easily replaceable because they get messed up. These soft jaws should generally be softer than the workpieces being clamped, so that the soft jaws do not mark the workpiece. The suggestion to use some exotic and extremely hard, dense wood is not well thought out.
Well, from my thoughts above, I'm not convinced on the absolute need to use a jaw that is softer than the workpiece. It is more important that the jaw is smooth (imperfections in the jaw face will emboss on the workpiece) and just the right amount of grippy. A grippy surface (not smooth metal, or ipe for example), will allow you to use a minimum clamping pressure. Too much clamping pressure will emboss the jaw edges on your workpiece, regardless of how hard or soft the jaw is.
I agree that I would rule out the harder hardwoods. Ipe, ironwood, ebony, among others would make lousy choices not much better than aluminum. High quality hard maple too would not be all that good a choice. But less expensive softer maples make excellent jaws.