That question is too general. Different snares, different materials would be ideal. A Squirrel stick will do best with loops made of metal because the loops can be placed in various positions and it will stay at a certain location. Making a deadfall and you might want heavier materials for the supports. For a pit trap you might want lighter materials. Generally though, I like copper wire. Holds a nice loop, flexible, but enough rigidness to stay where you place it.
I am partial to 18 gauge copper thermostat wire, 20 gauge runs a close second. Thin diameter and very strong, great for a running snare or a squirrel pole snare, could be used as a garrote as well. Plus, being in the HVAC trade, I have a ready supply at all times. It's not very expensive and you can purchase it by the foot at hardware stores. You have a choice of 2, 3, 5, 8 and 10 conductor wires so, a few feet of one of those and you have lots of wire.
Any of the micro-braid fishing lines would be a good choice as well as the heavier nylon monofilament lines.
Go down to your local music store / guitar shop, ask them nicely, and they will most likely save and give you old guitar strings as they change out strings on customers guitars and the guitars their working on. They work great for small game snares and for other things as well.