ABS is the name.

Of course, usual printing techniques give you a laminated object, as opposed to a homogeneous one, but I'm sure in no time someone will come up with a cool way to solve that.

Yeah, the material will probably be good, but the biggest issue with this approach is more to do with the size tolerances in production. Lego bricks are made to an insanely low tolerance of size variance (I've read it's as low as 2µm!) which I doubt any 3D printer could match (yet). It's this size tolerance that leads to the consistent feeling of bricks, and things like always being able to firmly hold an item in a minifigure's hand without it being loose, but still being able to remove it from the hand.

If somebody could come up with a low-cost, crowd-sourceable way of reproducing this process it would be amazing — there are so many things fans could come up with that Lego would never do officially, and third-parties end up charging a fortune for. I'd be beating a path to the door of some CAD designer to commission helmets and armour for Noble Team, for starters…