Lots of different materials will work well for a picnic blanket. If you go with a water resistant layer, keep in mind that you still want to pick as dry a spot as possible. Even a truly waterproof tarp will get small holes in it fairly quickly, so if you put it down on a soggy lawn, your bottom will get wet when you sit down.
Generally you'll want two layers, a waterproof(ish) bottom layer, and a comfortable cloth/blanket top layer. Depending on your preferences, you can buy a purpose-made picnic tarp that has the two layers sewn together, or you can get two separate layers and just lay them out one on top of the other. Having them sewn together is more convenient, but the whole thing will be more difficult to clean, and take longer to dry. Having them as separate layers lets you throw the cloth layer in the wash and dryer quite easily, and you can replace the waterproof layer whenever it gets too many holes in it.
Waterproof(ish) layer options:
- a normal blue or green tarp / tarpaulin, usually made from woven strips of polyethylene, such as you may already own for hauling leaves when gardening or covering a roof while the shingles are being replaced
- an old shower curtain (convenient, but not very classy; make sure it's not visibly moldy)
- plastic painters dropcloth (inexpensive and easy to replace, but not very durable)
- a plasticized tablecloth, one of the more durable kind that have sort of a feltlike fabric on the underside, not the single-use kind (unless you're picnicking on very short notice and can't find anything else; the disposable kind will rip pretty quickly so you won't get more than a couple of uses out of it)
- any old blanket or bedsheet that's about the same size as your tarp, or a little smaller