I picked up a rather nice brass lamp in a second-hand shop, but the
internal wiring was rather gruesome, so I've pulled it all out to rewire
from scratch.

The bulb contacts are riveted into a pair of ceramic holders; here's a
picture https://www.dropbox.com/s/2mscxp11nl3u4d3/IMG_0238.jpg?dl=0.

The wire is crimped into the contacts, in a very inaccessible way. I
think the only way of rewiring onto those contacts will be to drill out
the rivets, then rivet them back in again once I have replaced the wire,
unless anyone has any better suggestions.

Are those ceramic holders a standard kind of fitting?

The lamp can be adjusted for height, and a section of the wiring was
this curly cable:
https://www.dropbox.com/s/2mscxp11nl3u4d3/IMG_0238.jpg?dl=0. It's
two-core, and what looks like the third core in the photo is actually
elastic, to accommodate the stretch.

Does that kind of cable have a name?

Thanks,

Daniele