I would call it a "spring contact". Typically made of phosphor-bronze, etc.
Beyond the obvious specifications (dimensions, thickness, tolerances, material, etc.)
There are dozens of variable factors depending on your application. Like:
How "springy" does it need to be? (metal alloy, thickness, etc.)
What is it going to be bearing against? (copper on fiberglass PC board?, contact points molded into some custom plastic part?)
How many cycles do you expect for its life-span?
Does it need to be low-resistance? (like gold contact surfaces)
I'm sure there are dozens of other factors that I didn't think of at the moment.